The Persistent Topology of Optimal Transport Based Metric Thickenings

A metric thickening of a given metric space $X$ is any metric space admitting an isometric embedding of $X$. Thickenings have found use in applications of topology to data analysis, where one may approximate the shape of a dataset via the persistent homology of an increasing sequence of spaces. We introduce two new families of metric thickenings, the $p$-Vietoris-Rips and $p$-\v{C}ech metric thickenings for all $1\le p\le \infty$, which include all measures on $X$ whose $p$-diameter or $p$-radius is bounded from above, equipped with an optimal transport metric. The $p$-diameter (resp. $p$-radius) of a measure is a certain $\ell_p$ relaxation of the usual notion of diameter (resp. radius) of a subset of a metric space. These families recover the previously studied Vietoris-Rips and \v{C}ech metric thickenings when $p=\infty$. As our main contribution, we prove a stability theorem for the persistent homology of $p$-Vietoris-Rips and $p$-\v{C}ech metric thickenings, which is novel even in the case $p=\infty$. In the specific case $p=2$, we prove a Hausmann-type theorem for thickenings of manifolds, and we derive the complete list of homotopy types of the $2$-Vietoris-Rips thickenings of the $n$-sphere as the scale increases.


List of Symbols
The list below describes several symbols that will be used throughout the article.

Introduction
Geometric simplicial complexes, such as Vietoris-Rips orČech complexes, are one of the cornerstones of topological data analysis. One can approximate the shape of a dataset X by building a growing sequence of Vietoris-Rips complexes with X as the underlying set, and then computing persistent homology. The shape of the data, as measured by persistence, is reflective of important patterns within [Car09].
The popularity of Vietoris-Rips complexes relies on at least three facts. First, Vietoris-Rips filtrations and their persistent homology signatures are computable [Bau21]. Second, Vietoris-Rips persistent homology is stable [CCSG + 09,CdSO14], meaning that the topological data analysis pipeline is robust to certain types of noise. Third, Vietoris-Rips complexes are topologically faithful at low scale parameters: one can use them to recover the homotopy types [Lat01] or homology groups [CO08] of an unknown underlying space, when given only a finite noisy sampling.
At higher scale parameters, we mostly do not know how Vietoris-Rips complexes behave. This is despite the fact that one of the key insights of persistent homology is to allow the scale parameter to vary from small to large, tracking the lifetimes of features as the scale increases. Our practice is ahead of our theory in this regard: data science practitioners are building Vietoris-Rips complexes with scale parameters larger than those for which the reconstruction results of [Lat01,CO08] apply.
Stability implies that as more and more data points are sampled from some "true" underlying space M , the Vietoris-Rips persistent homology of the dataset X converges to the Vietoris-Rips persistent homology of M . The simplest possible case is when the dataset X is sampled from a manifold M , and so we cannot fully understand the Vietoris-Rips persistent homology of data without also understanding the Vietoris-Rips persistent homology of manifolds. However, not much is known about Vietoris-Rips complexes of manifolds, except at small scales [Hau95]. Even the Vietoris-Rips persistent homology of the n-sphere S n is almost entirely unknown.
Two potential obstacles for understanding the homotopy types of Vietoris-Rips complexes of a manifold M are: There are by now several strategies for handling these two obstacles. One strategy is to remain in the setting of Vietoris-Rips simplicial complexes. Obstacle (1) is then unavoidable. Regarding obstacle (2), Bestvina-Brady Morse theory has only been successfully applied at low scale parameters, allowing Zaremsky [Zar18] to prove that VR(S n ; r) recovers the homotopy type of S n for r small enough, but not to derive new homotopy types that appear as r increases. Simplicial techniques have been considered for a long time, but even successes such as an understanding of the homotopy types of the Vietoris-Rips complexes of the circle at all scales [AA17] are not accompanied by a broader Morse theory (though some techniques feel Morse-theoretic).
A second strategy is very recent. [LMO20,Oku19] show that the Vietoris-Rips simplicial complex filtration is equivalent to thickenings of the Kuratowski embedding into L ∞ (M ) or any other injective metric space: in particular, the two filtrations have the same persistent homology. This overcomes obstacle (1): the inclusion of a metric space into a thickening of its Kuratowski embedding is continuous, and indeed an isometry onto its image. This connection has created new opportunities, such as the Morse theoretic techniques employed by Katz [Kat83,Kat89,Kat91a,Kat91b]. This Morse theory allows one to prove the first new homotopy types that occur for Vietoris-Rips simplicial complexes of the circle and 2sphere, but have not yet inspired progress for larger scales, or for spheres above dimension two.
A third strategy is to consider Vietoris-Rips metric thickenings, which rely on optimal transport and Wasserstein-distances [AAF18]. We refer to these spaces as the ∞-metric thickenings, for reasons that will become clear in the following paragraph. Such thickenings were invented in order to enable Morse-theoretic proofs of the homotopy types of Vietoris-Rips type spaces. The first new homotopy type of the ∞-Vietoris-Rips metric thickening of the n-sphere is known [AAF18], but only for a single (non-persistent) scale parameter. It was previously only conjectured that the ∞-Vietoris-Rips metric thickenings have the same persistent homology as the more classical Vietoris-Rips simplicial complexes [AAF18, Conjecture 6.12]; one of our contributions is to answer this conjecture in the affirmative. [Mir20] considers a Morse theory in Wasserstein space, which is inspired in part by applications to ∞-Vietoris-Rips metric thickenings, but which does not apply as-is to these thickenings as the ∞-diameter functional is not "λ-convex" [San17].
We introduce a generalization: the p-Vietoris-Rips metric thickening for any 1 ≤ p ≤ ∞. Let X be an arbitrary metric space. For 1 ≤ p ≤ ∞, the p-Vietoris-Rips metric thickening at scale parameter r > 0 contains all probability measures on X whose p-diameter is less than r. The p-Vietoris-Rips metric thickening will be equipped with the topology induced from the weak topology on P X . When X is bounded, the weak topology is generated by an optimal transport based metric; see Corollary A.2. For p finite, the p-diameter of a probability measure α on the metric space X is defined as diam p (α) := X×X d p X (x, x ) α(dx) α(dx ) 1/p , and diam ∞ (α) is defined to be the diameter of the support of α. Figure 1: The p-Vietoris-Rips bifiltration VR p (X; r) for X a metric space of three points in R 2 , with P X visualized as the convex hull of X in R 2 . Note VR p (X; r) ⊆ VR p (X; r ) for r ≤ r and p ≥ p .
The p-Vietoris-Rips metric thickenings at scale r form a metric bifiltration of X that is covariant in r and contravariant in p. Indeed, we have an inclusion map VR p (X; r) VR p (X; r ) for r ≤ r and p ≥ p ; see Figure 1. As one of our main contributions, we prove that the p-Vietoris-Rips metric thickening is stable. This means that if two totally bounded metric spaces X and Y are close in the Gromov-Hausdorff distance, then their filtrations VR p (X; •) and VR p (Y ; •) are close in the homotopy interleaving distance. This was previously unknown even in the case p = ∞ (see [AAF18,Conjecture 6.14]); we prove stability for all 1 ≤ p ≤ ∞. As a consequence, it follows that the (undecorated) persistent homology diagrams for the Vietoris-Rips simplicial complexes VR(X; r) and for the p = ∞ Vietoris-Rips metric thickenings VR ∞ (X; •) are identical. In other words, the persistent homology barcodes for VR(X, •) and VR ∞ (X; •) are identical up to replacing closed interval endpoints with open endpoints, or vice-versa. This answers [AAF18, Conjecture 6.12] in the affirmative. Another consequence of stability is that the p-metric thickenings give the same persistence diagrams whether one considers all Radon probability measures, or instead the restricted setting of only measures with finite Figure 2: The p-Cech bifiltrationČ p (X; r) for X a metric space of three points in R 2 , with P X visualized as the convex hull of X in R 2 . NoteČ p (X; r) ⊆Č p (X; r ) for r ≤ r and p ≥ p . support.
The proof of stability for metric thickenings is more intricate than the proof of stability for the corresponding simplicial complexes (see for instance [CCSG + 09, CdSO14,Mém17]). Whereas simplicial complexes can be compared via simplicial maps, the direct analogues of simplicial maps on metric thickenings are not necessarily continuous. Thus, new techniques are required to construct continuous maps between metric thickenings. Our technique, relying on partitions of unity, allows us to continuously approximate measures on one metric space X by measures on another metric space Y (where this approximation depends on the Gromov-Hausdorff distance between X and Y ), thereby allowing us to construct the desired interleavings.
We furthermore introduce the p-Čech metric thickenings, again for 1 ≤ p ≤ ∞, and prove analogues of all of the above results. The p-Čech metric thickening at scale parameter r > 0 contains all probability measures supported on X whose p-radius is less than r; see Figure 2.
We also deduce the complete spectrum of homotopy types of 2-Vietoris-Rips and 2-Čech metric thickenings of the n-sphere, equipped with the Euclidean metric 2 : VR 2 ((S n , 2 ); r) first attains the homotopy type of S n for scales r ≤ √ 2, and then for all scales r > √ 2 the space is contractible. By contrast, the Vietoris-Rips simplicial complexes or the ∞-Vietoris-Rips metric thickenings of the n-sphere (with either the Euclidean or the geodesic metric) are only known for a bounded range of scales, including only a single change in homotopy type ([AAF18, Section 5]), even though infinitely many changes in homotopy type are conjectured (c.f. [ABF21, Question 8.1]). See however, [LMO20,Corollary 7.18] for results for round spheres with the ∞ metric.
One of our main motivations for introducing the p-Vietoris-Rips and p-Čech metric thickenings is to enable effective Morse theories on these types of spaces. The p-variance forČech metric thickenings is a minimum of linear functionals, and therefore fits in the framework of Morse theory for min-type functions [BBK14,GR97,Bry78,Mat82]. On the Vietoris-Rips side, we remark that gradient flows of functionals on Wasserstein space can be defined when the functional is "λ-convex" [San17,Mir20]. Though the ∞-diameter is not λ-convex, we hope that the p-diameter functional for p < ∞ may be λ-convex in certain settings.
Organization. In Section 2 we describe background material and set notation. We define the p-Vietoris-Rips and p-Čech metric thickenings in Section 3, and consider their basic properties in Section 4. In Section 5 we prove stability. We consider Hausmann-type theorems in Section 6, and deduce the 2-Vietoris-Rips metric thickenings of Euclidean spheres in Section 7. In Section 8 we bound the length of intervals in p-Vietoris-Rips and p-Čech metric thickenings using a generalization of the spread of a metric space, called the p-spread.
We conclude the paper by providing some discussion in Section 9. In Appendix A.1 we explain how the q-Wasserstein distance metrizes the weak topology for 1 ≤ q < ∞. We describe connections to min-type Morse theories in Appendix A.2. In Appendix A.3 we show that p-Čech thickenings of finite metric spaces are homotopy equivalent to simplicial complexes. We prove the persistent homology diagrams of the p-Vietoris-Rips and p-Čech metric thickenings of a family of discrete metric spaces in Appendix A.4, and we describe the 0-dimensional persistent homology of the p-Vietoris-Rips and p-Čech metric thickenings of an arbitrary metric space in Appendix A.5. We consider crushings in Appendix A.6. In Appendix A.7, we show that the main properties we prove for the (intrinsic) p-Čech metric thickening also hold for the ambient p-Čech metric thickening.

Background
This section introduces background material and notation.
Metric spaces and the Gromov-Hausdorff distance. Let (X, d X ) be a metric space. For any x ∈ X, we let B(x; r) := {y ∈ X | d X (x, y) < r} denote the open ball centered at x of radius r.
Given a metric space (X, d X ), the diameter of a non-empty subset A ⊂ X is the number diam(A) := sup a,a ∈A d X (a, a ), whereas its radius is the number rad(A) := inf x∈X sup a∈A d X (x, a). Note that in general we have Definition 2.1 (Uniform discrete metric space). For any natural number n, we use Z n to denote the metric space consisting of n points with all interpoint distances equal to 1.
Definition 2.2 (ε-net). Let X be a metric space. A subset U ⊂ X is called an ε-net of X if for any point x ∈ X, there is a point u ∈ U with d X (x, u) < ε.
Let X and Y be metric spaces. The distortion of an arbitrary function ϕ : The codistortion of a pair of arbitrary functions ϕ : We will use the following expression for the Gromov-Hausdorff distance between X and Y [KO99]: In particular, by letting y = ϕ(x) above, this means that See [BBI01,Chapter 7] for more details about the Gromov-Hausdorff distance.
Simplicial complexes. Two of the most commonly used methods for producing filtrations in applied topology are the Vietoris-Rips andČech simplicial complexes, defined as follows. Let X be a metric space, and let r ≥ 0. The Vietoris-Rips simplicial complex VR(X; r) has X as its vertex set, and contains a finite subset [x 0 , . . . , x k ] as a simplex when diam([x 0 , . . . , x k ]) := max i,j d X (x i , x j ) < r. TheČech simplicial complexČ(X; r) has X as its vertex set, and contains a finite subset [x 0 , . . . , x k ] as a simplex when ∩ k i=0 B(x i ; r) = ∅.
Probability measures and Wasserstein distances. Our main reference for measure theory elements is [Bog18] (albeit we use slightly different notation). Given a metric space (X, d X ), by P X we denote the set of all Radon probability measures on X. We equip P X with the weak topology: a sequence α 1 , α 2 , α 3 , . . . ∈ P X is said to converge weakly to α ∈ P X if for all bounded, continuous functions ϕ : X R, we have lim n ∞ X ϕ(x) α n (dx) = X ϕ(x) α(dx). The support supp(α) of a probability measure α ∈ P X is the largest closed set C such that every open set which has non-empty intersection with C has positive measure. If supp(α) consists of a finite set of points, then α is called finitely supported and can be written as α = i∈I a i · δ x i , where I is finite, a i ≥ 0 for all i, i∈I a i = 1, and each δ x i is a Dirac delta measure at x i . Let P fin X denote the set of all finitely supported Radon probability measures on X.
Given another metric space Y and a measurable map f : X Y , the pushforward map f : P X P Y induced by f is defined by f (α)(B) = α(f −1 (B)) for every Borel set B ⊂ Y . In the case of finitely supported probability measures, we have the explicit formula f i∈I a i · δ x i = i∈I a i ·δ f (x i ) , so f restricts to a function P fin X P fin Y . If f is a continuous map, then f is a continuous map between P X and P Y in the weak topology; see Chapter 5 in [Bog18]. In the finitely supported case, the restriction of f is a continuous map from P fin X to P fin Y . Given α, β ∈ P X , a coupling between them is any probability measure µ on X × X with marginals α and β: (π 1 ) µ = α and (π 2 ) µ = β, where π i : X × X X is the projection map defined by π i (x 1 , x 2 ) = x i for i = 1, 2. By Cpl(α, β) we denote the set of all couplings between α and β. Notice that Cpl(α, β) is always non-empty as the product measure α ⊗ β is in Cpl(α, β).
Given q ∈ [1, ∞), let P q,X be the subset of P X consisting of Radon probability measures with finite moments of order q, that is, measures α with X d q X (x, x 0 ) α(dx) < ∞ for some, and thus any, x 0 ∈ X. Note that when X is a bounded metric space, we have P q,X = P X . We can equip P q,X with the q-Wasserstein distance (or Kantorovich q-metric). In this setting, the q-Wasserstein distance is given by and it can be shown that d X W,q defines a metric on P q,X ; see Chapter 3.3 in [Bog18]. If q = ∞, then again for any metric space (X, d X ) we define the ∞-Wasserstein distance for α, β ∈ P ∞,X , the set of measures with bounded support. The ∞-Wasserstein distance d X W,∞ is clearly symmetric, and the triangle inequality comes from a similar gluing trick as in Chapter 3.3 of [Bog18]. Thus, d X W,∞ is a pseudo-metric that is lower bounded by a metric, for example d X W,1 ; therefore d X W,∞ is also a metric. For general q ≤ q , it follows from Hölder's inequality that d X W,q ≤ d X W,q . It is known that on a bounded metric space, for any q ∈ [1, ∞), the q-Wasserstein metric generates the weak topology. For a summary of the result, see Appendix A.1. On the other hand, the ∞-Wasserstein metric d X W,∞ generates a finer topology than the weak topology in general.
Let f, g : X R be measurable functions over the metric space X, and let p ∈ [1, ∞). We will frequently use the Minkowski inequality, which states that The following proposition shows that we may construct linear homotopies in spaces of probability measures, analogous to linear homotopies in Euclidean spaces. Linear homotopies will play an important role in Section 5.
Proposition 2.4. Let Z be a metric space (or more generally a first-countable space), let X be a metric space, and let f, g : Z P X be continuous. Then the linear homotopy H : Z × [0, 1] P X given by H(z, t) = (1 − t) f (z) + t g(z) is continuous.
Proof. It suffices to show sequential continuity, because Z × I is metrizable (or more generally first-countable) and the weak topology on the set P X is also metrizable; see [Bog18, Theorem 3.1.4] and also Appendix A.1. If {(z n , t n )} n converges to (z, t) in Z × I, then to show weak convergence of the image in P X , let γ : X R be any bounded, continuous function. We have where the second equality uses the fact that f (z n ) and g(z n ) are weakly convergent, since f and g are continuous. Therefore H(z n , t n ) converges weakly to H(z, t), so H is continuous.
Fréchet means. For each p ∈ [1, ∞] let the p-Fréchet function of α ∈ P X , namely F α,p : X R ∪ {∞}, be defined by Note that F α,p (x) = d X W,p (δ x , α). A point x ∈ X that minimizes F α,p (x) is called a Fréchet mean of α; in general Fréchet means need not be unique. See [Kar77] for some of the basic properties of Fréchet means.
Metric thickenings with p = ∞. Let X be a bounded metric space. The Vietoris-Rips andČech metric thickenings were introduced in [AAF18] with the notation VR m (X; r) andČ m (X; r), where the superscript m denoted "metric." We instead denote these spaces by VR fin ∞ (X; r) andČ fin ∞ (X; r), since one of the main contributions of this paper will be to introduce the generalizations VR p (X; r) andČ p (X; r) (and their finitely supported variants VR fin p (X; r) andČ fin p (X; r)) for any 1 ≤ p ≤ ∞. The Vietoris-Rips metric thickening VR fin ∞ (X; r) is the space of all finitely supported probability measures of the form k i=0 a i δ x i , such that diam({x 0 , . . . , x k }) < r, equipped with the q-Wasserstein metric for some 1 ≤ q < ∞. The choice of q ∈ [1, ∞) does not affect the homeomorphism type by Corollary A.2. TheČech metric thickeningČ fin ∞ (X; r) is the space of all finite probability measures of the form k i=0 a i δ x i , such that ∩ k i=0 B(x i ; r) = ∅, equipped with the q-Wasserstein metric for some 1 ≤ q < ∞.
Comparisons. We give a brief survey of the various advantages and disadvantages of using simplicial complexes and p = ∞ metric thickenings.
Despite this rich array of simplicial techniques, there are some key disadvantages to Vietoris-Rips simplicial complexes. If X is not a discrete metric space, then the inclusion from X into VR(X; r) for any r ≥ 0 is not continuous since the vertex set of a simplicial complex is equipped with the discrete topology. Another disadvantage is that even though we start with a metric space X, the Vietoris-Rips simplicial complex VR(X; r) may not be metrizable. Indeed, a simplicial complex is metrizable if and only if it is locally finite [Sak13, Proposition 4.2.16(2)], i.e. if and only if each vertex is contained in only a finite number of simplices. So when X is infinite, VR(X; r) is often not metrizable, i.e. its topology cannot be induced by any metric. Though Vietoris-Rips complexes accept metric spaces as input, they do not remain in this same category, and may produce as output topological spaces that cannot be equipped with any metric structure.
By contrast, the metric thickening VR fin ∞ (X; r) is always a metric space that admits an isometric embedding X VR fin ∞ (X; r) [AAF18]. Furthermore, metric thickenings allow for nicer proofs of Hausmann's theorem. For M a Riemannian manifold and r > 0 sufficiently small depending on the curvature of M , Hausmann produces a map T : VR(M ; r) M from the simplicial complex to the manifold that is not canonical in the sense that it depends on a total order of all points in the manifold. Also, the inclusion map M VR(M ; r) is not continuous, and therefore cannot be a homotopy inverse for T . Nevertheless, Hausmann is able to prove T is a homotopy equivalence without constructing an explicit inverse. By contrast, in the context of metric thickenings, one can produce a canonical map VR fin ∞ (M ; r) M by mapping a measure to its Fréchet mean (whenever r is small enough so that measures of diameter less than r have unique Fréchet means). The (now continuous) inclusion M VR fin ∞ (M ; r) can be shown to be a homotopy inverse via linear homotopies [AAF18, Theorem 4.2].
One of the main contributions of this paper is showing how these various spaces relate to each other, especially when it comes to persistent homology. This allows one to work either simplicially, geometrically, or with measures -whichever perspective is most convenient for the task at hand.
Homology and Persistent homology. For each integer k ≥ 0, let H k denote the singular homology functor from the category Top of topological spaces to the category Vec of vector spaces and linear transformations. We use coefficients in a fixed field, so that homology groups are furthermore vector spaces. For background on persistent homology, we refer the reader to [EH10,ELZ00,ZC05].
In applications of topology, such as topological data analysis [Car09], one often models a dataset not as a single space X, but instead as an increasing sequence of spaces. We refer to an increasing sequence of spaces, i.e. a functor from the poset (R, ≤) to Top, as a filtration. If X is a metric space, then a common filtration is the Vietoris-Rips simplicial complex filtration VR(X, •). In this paper we will introduce relaxed versions, the p-Vietoris-Rips metric thickening filtrations VR p (X; •) (see Section 3).
By applying homology (with coefficients in a field) to a filtration, we obtain a functor from the poset (R, ≤) to Vec, i.e. a persistence module. We will use symbols like V , W and so on to denote persistence modules. Following [CdSGO16], a persistence module is Q-tame if for any s < t, the structure map V (s) V (t) is of finite rank. In [CdSGO16], it is shown that a Qtame persistence module V can be converted into a persistence diagram 1 , dgm(V ), which is a multiset in the extended open half-plane consisting of pairs The persistence diagrams can be compared via the bottleneck distance d B , which is defined as follows. Given persistence diagrams D 1 and D 2 , a subset M ⊂ D 1 × D 2 is said to be a partial matching between D 1 and D 2 if the following are satisfied: • For every point p in D 1 , there is at most one point q in D 2 such that (p, q) ∈ M . (If there is no such q, we then say p is unmatched.) • For every point q in D 2 , there is at most one point p in D 1 such that (p, q) ∈ M . (If there is no such p, we then say q is unmatched.) The bottleneck distance d B between two persistence diagrams D 1 and D 2 is where s is an element in D 1 or D 2 , s b is the birth time of s, s d is the death time of s, and M varies among all possible partial matchings.
Interleavings Let C be a category. We call any functor from the poset (R, ≤) to C an Rspace. Such a functor gives a structure map X(s) X(t) for any s ≤ t. For X an R-space, the δ-shift of X is the functor X δ : R C with X δ (t) = X(t + δ) for all t ∈ R. We have a natural transformation id δ X from X to X δ which maps X(t) to X δ (t) using the structure maps from X. For two R-spaces X and Y , we say they are δ-interleaved if there are natural transformations F : X Y δ and G : Y X δ such that F • G = id 2δ X and G • F = id 2δ Y . Now we define a pseudo-distance between R-spaces, which we call the interleaving distance d C I , as follows: d C I (X, Y ) := inf{δ | X and Y are δ-interleaved}. We note that d Vec I is the interleaving distance for persistence modules. The isometry theorem of [Les15,CdSGO16] states that d Vec I (V, W ) = d B (dgm(V ), dgm(W )) for Q-tame persistence modules V and W .
We will use the following lemma in our proofs in Section 5 on stability.
Lemma 2.5. If a persistence module P can be approximated arbitrarily well in the interleaving distance by Q-tame persistence modules, then P is also Q-tame.
Proof. For any s < t, let ε = t − s, so there is a Q-tame persistence module P ε such that d I (P, P ε ) < ε 3 . Then using the maps of an ε 3 -interleaving between P and P ε , the structure map P (s) P (t) can be factored as follows.
As P ε is a Q-tame module, the rank of P ε s + ε 3 P ε (s + 2ε 3 ) is finite, and hence so is the rank of P (s) P (t). Since s and t are arbitrary, we get the Q-tameness of P .
The homotopy type distance We next recall the definition of the homotopy type distance d HT distance from [FLM19]. The following is a small generalization of [FLM19, Definition 2.2] in that we do not require the maps ϕ X and ϕ Y to be continuous.
Definition 2.6. Let (X, ϕ X ) and (Y, ϕ Y ) be two topological spaces with real-valued functions on them. For any δ ≥ 0, a δ-map between (X, ϕ X ) and (Y, ϕ Y ) is a continuous map Φ : X Y such that ϕ Y • Φ(x) ≤ ϕ X (x) + δ for any x ∈ X. For any two δ-maps Φ 0 : X Y and Φ 1 : X Y , a δ-homotopy between Φ 0 and Φ 1 with respect to the pair (ϕ X , ϕ Y ) is a continuous map H : Definition 2.7 ([FLM19, Definitions 2.5 and 2.6]). For every δ ≥ 0 and for any two pairs (X, ϕ X ) and (Y, ϕ Y ), we say (X, ϕ X ) and (Y, ϕ Y ) are δ-homotopy equivalent if there exist δ-maps Φ : X Y and Ψ : Y X, with respect to (ϕ X , ϕ Y ) and (ϕ Y , ϕ X ), respectively, such that • the map Ψ • Φ : X X is 2δ-homotopic to id X with respect to (ϕ X , ϕ X ), and If (X, ϕ X ) and (Y, ϕ Y ) are not δ-homotopy equivalent for any δ, then we declare Proposition 2.8 ([FLM19, Proposition 2.10]). The d HT distance is an extended pseudometric on the set of pairs of topological spaces and real-valued functions.
A pair (X, ϕ X ) (where X is a topological space and ϕ X : X R is a real-valued function) induces an R-space [X, ϕ X ; •] given by the sublevel set filtration, and defined as follows: The following theorem shows the interleaving distance of the persistent homology of the sublevel set filtrations is bounded by the d HT distance of the respective pairs. The theorem is a slight generalization of [FLM19, Lemma 3.1] in which we do not require the continuity of ϕ X and ϕ Y ; we omit its (identical) proof.

The p-relaxation of metric thickenings
We now describe the construction of the p-relaxations of metric thickenings.

The relaxed diameter and radius functionals
Throughout this section, (X, d X ) will denote a bounded metric space. Recall that P X is the set of all Radon probability measures on X, equipped with the weak topology. We now introduce for each p ∈ [1, ∞] the p-diameter of a measure α in P X , where the ∞-diameter of α is precisely the diameter of its support supp(α) (as a subset of X). Consider for each p ∈ [1, ∞] the p-diameter map diam p : P X R ≥0 given by We remark that the p-diameter has been studied and critiqued as a measure of diversity in population biology [Rao82,POP05], and it has also been considered in relation to the Gromov-Wasserstein distance [Mém11]. Similarly, define the p-radius map rad p : Note that the p-radius of a measure is simply the p-th root of its p-variance. We observe that Then we have Since ε > 0 was arbitrary, this shows diam p (α) ≤ 2 rad p (α).
Remark 3.2. The tightness of the bound rad p (α) ≤ diam p (α) can be seen from the calculation for the uniform measure on Z n , the metric space on a set of size n with all interpoint distances equal to 1. To see the (asymptotic) tightness of diam p (α) ≤ 2 rad p (α), we consider the metric space Z n ∪ {O}, where the newly introduced "center" O has distance 1/2 to every other point. Then for the measure α := z∈Zn 1 n δ z , we have rad p (α) = 1/2 and diam p (α) = ( n−1 n ) 1 p . We obtain asymptotic tightness by letting n go to infinity.

The relaxed Vietoris-Rips andČech metric thickenings
We regard VR p (X; r) as a topological space by endowing it with the subspace topology from P X (i.e., the weak topology). By convention, when r ≤ 0, we will let VR p (X; r) = ∅. By we will denote the filtration thus induced.
We use VR fin p (X; r) and VR fin p (X; •) to denote the finitely-supported variants, obtained by replacing P X in the above definition with P fin X .
Note that in the specific case p = ∞, the ∞-diameter of a measure is simply the diameter of its support. For a bounded metric space X, the weak topology is generated by the 1-Wasserstein metric, so the definition of VR fin p (X; r) generalizes [AAF18, Definition 3.1], which is the specific case p = ∞.
Remark 3.4 (VR p (X; •) as a softening of VR ∞ (X, •)). We should point out that the definition of VR p (X; •) can be extended to the whole range p ∈ [0, ∞] as follows. One first notices that diam p (α) can still be defined as above for p ∈ (0, 1), and by diam 0 (α) = 0 when which equals 0 since the product contains terms with i = j. At any rate, by the standard generalized means inequality [BMV13], we have diam p (α) ≤ diam p (α) for any p ≤ p in the range [0, ∞]. So for fixed r > 0, not only does VR p (X; r) become larger and larger as p decreases, but also for p = 0, VR 0 (X; r) contains all Radon probability measures on X and thus has trivial reduced homology.
Definition 3.5 (p-Čech filtration). For each r > 0 and p ∈ [1, ∞], let the p-Čech metric thickening at scale r beČ We regardČ p (r; X) as a topological space by endowing it with the subspace topology from P X (i.e., the weak topology). By convention, when r ≤ 0, we will letČ p (X; r) = ∅. ByČ we will denote the filtration thus induced. 2 Remark 3.6. Let α be a measure in P X . As rad p (α) = inf x∈X d X W,p (δ x , α), we have rad p (α) < r if and only if there is some x ∈ X such that d X W,p (δ x , α) < r. Therefore,Č p (X; r) is exactly the union over all x ∈ X of the balls B(δ x ; r), with respect to the p-Wasserstein metric, centered at points δ x in the isometric image of X in P X .
We useČ fin p (X; r) andČ fin p (X; •) to denote the finitely-supported variants, obtained by replacing P X in the above definition with P fin X . Note that in the specific case p = ∞, the ∞-radius of a measure is simply the radius of its support. For a bounded metric space X, the weak topology is generated by the 1-Wasserstein metric, so the definition ofČ fin p (X; r) generalizes [AAF18] which considers the specific case p = ∞.
Though the above definitions are given with the < convention, we remark that they instead could have been given with the ≤ convention, namely diam p (α) ≤ r or rad p (α) ≤ r. In this paper, we restrict attention to the < convention, even though many of the statements we give are also true with the ≤ convention.
The next proposition shows that p-Vietoris-Rips and p-Čech metric thickenings are nested in the same way that Vietoris-Rips andČech simplicial complexes are.
Proof. This follows immediately from Proposition 3.1, which implies that for any α ∈ P X , If X and Z are metric spaces with X ⊆ Z, and if the metric on Z is an extension of that on X, then following Gromov [Gro93, Section 1B] we say that Z is a r-metric thickening of Proposition 3.8. Let X be a bounded metric space. When equipped with the q-Wasserstein metric for 1 ≤ q ≤ p, we have thatČ p (X; r) and VR p (X; r) are each r-metric thickenings of X for all r > 0.
Proof. We use the isometric embedding X Č p (X; r) given by , which shows thatČ p (X; r) equipped with the q-Wasserstein metric is an r-metric thickening of X.
The Vietoris-Rips case follows immediately since VR p (X; r) ⊆Č p (X; r) by Proposition 3.7.
The following remark follows the perspective introduced in the recent paper [LMO20], which shows that the filling radius is related to the persistent homology of the Vietoris-Rips simplicial complex filtration VR(X, •).
Remark 3.9. Let X be a closed connected n-dimensional manifold so that the fundamental class of X is well-defined. Let Z be an r-metric thickening of X for some r > 0. It is shown in [Gro83, Page 8, Another Corollary] that the map induced from the inclusion ι : X Z is an injection whenever r is less than a scalar geometric invariant FillRad(X) called the filling radius of X. Hence Proposition 3.8 implies that for all p ∈ [1, ∞], the persistence diagram in dimension n of either the filtration VR p (X; •) orČ p (X; •) contains an interval that with left endpoint equal to 0 and length at least FillRad(X). As proved in [Kat83], the filling radius of the n-sphere (with its geodesic metric) is FillRad(S n ) = 1 2 arccos −1 n+1 . Therefore, when X = S n , the n-dimensional persistence diagram of either VR p (S n ; •) orČ p (S n ; •) contains an interval starting at zero which is no shorter than 1 2 arccos −1 n+1 . Lemma 3.10. For all r > 0 and all p, p ∈ [1, ∞] with p ≥ p , one has VR p (X; r) ⊆ VR p (X; r) andČ p (X; r) ⊆Č p (X; r). Proof. This comes from applying the Hölder inequality on diam p and rad p .
Example 3.11. We recall that Z n+1 is the metric space consisting of n + 1 points with all interpoint distances equal to 1. For any natural number n, the Vietoris-Rips orČech simplicial complex filtrations of Z n+1 do not produce any non-diagonal point in their persistence diagram except in homological dimension zero. However, for any 1 < p < ∞, both the p-Vietoris-Rips and p-Čech filtrations of Z n+1 will contain non-diagonal points in their persistence diagrams for homological degrees from 0 to n − 1. More specifically, both the p-Vietoris-Rips and p-Čech filtrations are homotopy equivalent to a filtration of an n-simplex ∆ n by its k-skeleta, ∆ (k) n . We prove the following result in Appendix A.4. For r in the interval and when r > n n+1 1 p , both VR p (Z n+1 ; r) andČ p (Z n+1 ; r) become the n-simplex ∆ n which is contractible.
From this we get the persistence diagrams

A more general setting: controlled invariants
We now generalize the relaxed p-Vietoris-Rips and p-Čech filtrations.
Definition 3.12 (Controlled invariants on P X ). Let i be a functional that associates to each bounded metric space X a function i X : P X R. We say, for some L > 0, that i is an L-controlled invariant if the following conditions are satisfied: 1. Stability under pushforward : For any map f between finite metric spaces X and Y and any α in P X , 2. Stability with respect to d X W,∞ : For any bounded metric space X and any α, β in P X , In the next section, we will prove that both diam p and rad p induce controlled invariants. For any controlled invariant i, we will use the notation [P X , i X ; •] to denote the sublevel set filtration induced by the pair (P X , i X ). That is, This construction generalizes both the definition of VR p (X; •) and that ofČ p (X; •). Similarly, any controlled invariant i : P fin X R induces an analogous sublevel set filtration [P fin We'll see later on in Corollary 5.3 that the filtrations [P X , i X ; •] and [P fin X , i X ; •] yield persistence modules which are -interleaved for arbitrary small > 0.

Basic properties
We prove some basic properties. These include the convexity of the Wasserstein distance, the fact that diam p and rad p are 1-controlled, and the fact that for X finite, the p = ∞ metric thickenings VR ∞ (X; r) andČ ∞ (X; r) are homeomorphic to the simplicial complexes VR(X; r) andČ(X; r), respectively. We begin with the property that nearby measures have nearby integrals. Throughout this section, X is a bounded metric space.
Lemma 4.1. Let ϕ be an L-Lipschitz function on X. Then for any α, β in P X and any Proof. Let µ ∈ Cpl(α, β) be any coupling. Then we have We obtain the claim by infimizing the right hand side with respect to the coupling µ.
We will often use the following convexity result of the Wasserstein distance; for a more general result, see [Vil08, Theorem 4.8].
Lemma 4.2. If c 1 , . . . , c n are non-negative real numbers satisfying n k=1 c k = 1 and if α 1 , . . . , α n , α 1 , . . . , α n ∈ P X , then for all q ∈ [1, ∞), we have Proof. Let ε > 0 and q ∈ [1, ∞). For each k, suppose µ k is a coupling between α k and α k such that Then it can be checked that n k=1 c k µ k is a coupling between n k=1 c k α k and n k=1 c k α k . We have Since this holds for all ε > 0, the claimed inequality holds. The case for q = ∞ can be checked separately using the same matching.
We define an invariant generalizing diam p , and then prove that it is 1-controlled.
Definition 4.3 (i X q,p invariant). Let X be a bounded metric space. For any p, q ∈ [1, ∞], we define the i q,p invariant, which associates to each bounded metric space X a function i X q,p : P X R, Remark 4.4. In the contruction of i X q,p , one can get other interesting filtration functions by replacing the Wasserstein distance with other distances between measures. These include, for example, the Lévy-Prokhorov metric, the Fortet-Mourier metric, and variants of the Kantorovich-Rubinstein metric. For definitions of these metrics, see [Bog18, Section 3].
Lemma 4.5. Let X and Y be bounded metric spaces, and let f : X Y be a map. Then for any element α ∈ P X and p ∈ [1, ∞], we have Lemma 4.6. Let X be a bounded metric space. Then for any α, β in P X , we have In particular, |i X q, Proof. We first note that We also have where the last line follows from Lemma 4.
. We then get the result by swapping the roles of α and β.
Remark 4.7. This establishes the continuity of i q,p for q, p ∈ [1, ∞). If p or q equals infinity, then i q,p is not necessarily continuous in the weak topology since d X W,∞ does not necessarily induce the same topology on P X as d X W,p does for p < ∞. Indeed, if X = [0, 1] is the unit interval, then n−1 n δ 0 + 1 n δ 1 δ 0 in P X as n ∞, even though diam ∞ ( n−1 n δ 0 + 1 n δ 1 ) is equal to 1 for all n ≥ 1, and hence does not converge to 0 = diam(δ 0 ).
The above two lemmas imply that i q,p (and hence diam p = i p.p ) is a 1-controlled invariant. We next consider the analogous properties for rad p .
Lemma 4.8 (rad p under a map with bounded distortion). Let X and Y be bounded metric spaces, and let f : X Y be a map. Then for any α ∈ P X and p ∈ [1, ∞], we have Proof. We only give the proof for the case p < ∞; the case p = ∞ is similar.
Lemma 4.9 (Stability of rad p ). Let X be a bounded metric space. For any two probability measures α, β ∈ P X and for every p ∈ [1, ∞], we have Proof. We compute Remark 4.10. Lemma 4.9 establishes the continuity of rad p for p ∈ [1, ∞), as these functions are 1-Lipschitz. Similarly to diam ∞ , we note that rad ∞ is not necessarily continuous because the metric topology given by d X W,∞ is not necessarily equal to the weak topology. The above two lemmas imply that rad p is a 1-controlled invariant. We end this section of basic properties by showing that P X is homeomorphic to a simplex when X is finite. Hence for X finite the p = ∞ metric thickenings VR ∞ (X; r) anď C ∞ (X; r) are homeomorphic to the simplicial complexes VR(X; r) andČ(X; r), respectively (see also [AAF18,Corollary 6.4]).
Lemma 4.11. If X is a finite metric space with n points, then P X is homeomorphic to the standard (n − 1)-simplex.
Proof. Let X = {x 1 , . . . , x n }. The space P X of probability measures is in bijection with the standard n − 1 simplex ∆ n−1 = {(y 1 , . . . , y n ) ∈ R n | n i=1 y i = 1, y i ≥ 0 for all i} via the function f : P X ∆ n−1 defined by f ( n i=1 a i δ x i ) = (a 1 , . . . , a n ). Suppose we have a sequence {α k } in P X given by α k = n i=1 a k,i δ x i . By the definition of weak convergence, {α k } converges to α = n i=1 a i δ x i in P X if and only if X φ(x)α k (dx) converges to X φ(x)α(dx) for all bounded and continuous functions φ : X R. These integrals are equal to n i=1 a k,i φ(x i ) and n i=1 a i φ(x i ) respectively, so {α k } converges to α if and only if lim k ∞ a k,i = a i for each i. Therefore, {α k } converges to α in P X if and only if {f (α k )} converges to f (α) in ∆ n−1 .
In general, the p = ∞ metric thickenings VR ∞ (X; r) andČ ∞ (X; r) of a finite metric space X are in bijection, as sets, with the geometric realizations of the usual Vietoris-Rips andČech simplicial complexes on X, via the natural bijection a i x i denotes the formal sum in the geometric realization. Therefore, Lemma 4.11 implies the following; see also [AAF18, Proposition 6.6] Lemma 4.12. For a finite metric space X and any r > 0, we have homeomorphisms VR ∞ (X; r) ∼ = VR(X; r) andČ ∞ (X; r) ∼ =Č(X; r).

Stability
In this section we establish the stability of all the filtrations we have introduced in this paper, under reasonable assumptions on the metric spaces. The proof requires new techniques in order to construct maps that adequately compare two filtrations of metric thickenings. In the proof of stability for simplicial complexes ( [CdSO14]), simplicial maps are used to construct maps between filtrations of simplicial complexes, and the shift in parameter can be bounded using the Gromov-Hausdorff distance. A naive attempt to apply this technique to filtrations of metric thickenings fails because there is no analogue of simplicial maps for metric thickenings. Indeed, if X and Y are metric spaces that are close in the Gromov-Hausdorff distance, then we get a map f : X Y that need not be continuous, but which is of bounded distortion. This induces a continuous map VR(X; r) VR(Y ; r ) between simplicial complexes, as long as r is chosen to be large enough, and from there one can obtain interleavings. However, the analogous map VR p (X; r) VR p (Y ; r ) between metric thickenings cannot be continuous if f : X Y is not continuous, since there are natural isometric embeddings of X and Y into VR p (X; r) and VR p (Y ; r ) (see Proposition 3.8). In essence, the fact that metric thickenings have a more well-behaved topology means that it is more difficult to construct interleavings between them.
To overcome this difficulty, we instead construct continuous functions between metric thickenings by distributing mass according to finite partitions of unity, subordinate to open coverings by δ-balls. This ensures that these maps distort distances in a controlled way, allowing for a controlled change in the invariants defining the filtration and thus providing an interleaving. Creating these maps and checking their properties will require many of the ideas from the previous sections.
In Section 5.1 we state our main results, Theorem A and its immediate consequence, Theorem B. We give two applications of stability. First, Section 5.2 applies the stability theorem in order to show that various persistence modules of interest are Q-tame, and therefore have persistence diagrams. Next, in Section 5.3 we apply stability to show the close relationship between ∞-metric thickenings and simplicial complexes, generalizing and answering in the affirmative [AAF18, Conjecture 6.12] which states that these filtrations have identical persistence diagrams. We give the proof of the stability theorem, Theorem A, in Section 5.4.

Statement of the stability theorem
Let X and Y be totally bounded metric spaces, let p ∈ [1, ∞], and let k ≥ 0. We will see, for example, that we have the following stability bound: This implies that if two metric spaces are close in the Gromov-Hausdorff distance, then their resulting filtrations and persistence modules are also close.
Example 5.1. In this example we consider the metric space Z n+1 from Example 3.11, which has n + 1 points all at interpoint distance 1 apart. Let ζ p,k be the interleaving distance between persistent homology modules Notice that from Example 3.11, when n = m and p is finite, we have However, when p = ∞, ζ ∞,0 = 1 2 and ζ ∞,k = 0 for all k ≥ 1. From these calculations we can make the following observations: (i) For p infinite, the only value of k for which ζ ∞,k = 0 is k = 0.
From items (i) and (ii) above we can see that whereas persistence diagrams for k ≥ 1 of p = ∞ thickenings do not contain discriminative information for the Z n spaces, in contrast, the analogous quantities for p finite do absorb useful information.
Since it is known (cf. [Mém12,Example 4.1]) that d GH (Z n+1 , Z m+1 ) = 1 2 whenever n = m, item (iii) above suggests that the lower bound for Gromov-Hausdorff given by Equation (4) may not be tight for p finite. This phenomenon can actually be explained by the more general theorem below, which identifies a certain pseudo-metric between filtrations which mediates between the two terms appearing in Equation (4).
Theorem A. Let i be an L-controlled invariant. Then for any two totally bounded metric spaces X and Y and any integer k ≥ 0, we have Note that there are instances when the quantity in the middle vanishes yet the spaces X and Y are non-isometric; see Appendix A.6 for results about this in terms of the notion of crushing considered by Hausmann [Hau95] and [MO21,AAF18].
Corollary 5.2. For any two totally bounded metric spaces X and Y , for any q, p ∈ [1, ∞], and for any integer k ≥ 0, we have Proof. From Lemma 4.5 and Lemma 4.6 we know i q,p is a 1-controlled invariant, and then we apply Theorem A.
Theorem B. Let X and Y be totally bounded metric spaces, let p ∈ [1, ∞], and let k ≥ 0 be an integer. Then the k-th persistent homology of VR p (X; •) and VR p (Y ; •) are ε-interleaved for any ε ≥ d GH (X, Y ), and similarly forČ p (X; •) andČ p (Y ; •): Proof. The p-Vietoris-Rips case follows from Corollary 5.2 by letting q = p, in which case i X p,p = diam X p (see Definition 4.3). The p-Čech case follows from Theorem A since rad p is a 1-controlled invariant by Lemmas 4.8 and 4.9.
In the corollary below, we use Theorem A to show that a controlled invariant on all Radon probability measures produces a filtration at interleaving distance zero from that same invariant restricted to only finitely supported measures.
Corollary 5.3. Let X be a totally bounded metric space, let k ≥ 0 be an integer, and let i be an L-controlled invariant. Then Proof. For any ε > 0, let U be a ε-net in X. Then the stability theorem, Theorem A, shows Since d HT satisfies the triangle inequality (Proposition 2.8), we have d HT (P X , i X ), (P fin X , i X ) ≤ 4Lε. By letting ε go to zero, we see d HT (P X , i X ), (P fin X , i X ) = 0. It then follows from Lemma 2.9 that

Consequence of stability: Tameness
We next determine conditions under which a persistence module H k • [P X , i X ; •] associated with a controlled invariant i will be Q-tame. In particular, we apply the results to the Vietoris-Rips andČech metric thickenings.
Corollary 5.4. Let i be a controlled invariant and let k ≥ 0 be an integer. Suppose the is Q-tame for any finite metric space V . Then for any totally bounded metric space X, the persistence module Proof. Since X is totally bounded, Theorem A implies that the persistence module H k • [P X , i X ; •] can be approximated arbitrarily well in the interleaving distance by the Q-tame persistence modules on finite ε-nets V ε as ε goes to zero. Then the result follows from Lemma 2.5. Now, we give a sufficient condition for the Q-tameness of the persistence module H k • [P V , i V ; •] over a finite metric space V .
Corollary 5.5. Suppose i is a controlled invariant such that for any finite metric space V , i V is a continuous function on P V . Then for any totally bounded metric space X, the persistence modules H k • [P X , i X ; •] and H k • [P fin X , i X ; •] are Q-tame for any integer k ≥ 0.
Proof. According to Corollary 5.4, it suffices to check Q-tameness over finite metric spaces. If V is a finite metric space, then we may identify P V with a simplex by Lemma 4.11. Then [P V , i V ; •] is a sublevel set filtration on a simplex, and since i V is continuous, Theorem 2.22 of [CdSGO16] shows Here we summarize the Q-tameness results related to relaxed Vietoris-Rips andČech metric thickenings. Proof. For H k •VR p (X; •), when p is finite, Lemma 4.6 implies diam p is a continuous function over any P X where X is a bounded metric space. Then we get the result by applying Corollary 5.5. When p = ∞, Lemma 4.12 shows the persistent homology associated with diam ∞ is Q-tame over finite metric spaces. We then get the result by applying Corollary 5.4. We obtain the case of H k • VR fin p (X; •) from H k • VR p (X; •) by using the interleaving distance result in Corollary 5.3 along with the Q-tame approximation result in Lemma 2.5.
For H k •Č p (X; •), when p is finite, Lemma 4.9 implies rad p is a continuous function over any P X where X is a bounded metric space. The rest is similar to the Vietoris-Rips case.
Remark 5.7. The proof that H k •Č p (X; •) is Q-tame can be made more direct at one step. For U finite, to see that H k •Č p (U ; •) is Q-tame, one could appeal to Theorem F in Appendix A.3 instead of Theorem 2.22 of [CdSGO16].
The Q-tame persistence modules given by this theorem allow us to discuss persistence diagrams, using the results of [CdSGO16].
Corollary 5.8. Let X be a totally bounded metric space, let p ∈ [1, ∞], and let k ≥ 0 be an integer. Then H k • VR p (X; •) and H k • VR fin p (X; •) have the same persistence diagram, denoted dgm VR k,p (X). Similarly, H k •Č p (X; •) and H k •Č fin p (X; •) have the same persistence diagram, denoted dgmČ k,p (X) Proof. Persistence diagrams are well-defined for Q-tame persistence modules, so for any totally bounded metric space X, any p ∈ [1, ∞], and any k ≥ 0, by Corollary 5.6 we have persistence diagrams associated to H k • VR p (X; •) and H k • VR fin p (X; •). From Corollary 5.3 we know that the interleaving distance between H k • VR p (X; •) and H k • VR fin p (X; •) is zero, and so the Isometry Theorem [CdSGO16, Theorem 4.11] implies that these persistence modules have the same (undecorated) persistence diagram, denoted dgm VR k,p (X). The same proof also works forČech metric thickenings.
Combining the Isometry Theorem ([CdSGO16, Theorem 4.11]) with Theorem B and Corollary 5.6, we obtain the following.
Corollary 5.9. If X and Y are totally bounded metric spaces, then for any p ∈ [1, ∞] and any integer k ≥ 0, we have
We recall that VR p (X; r) denotes the p-Vietoris-Rips metric thickening, that VR fin p (X; r) denotes the p-Vietoris-Rips metric thickening for measures of finite support, and that VR(X; r) denotes the Vietoris-Rips simplicial complex. Theorem B shows, for any p ∈ [1, ∞], any δ > 0, and any finite δ-net U δ of a totally bounded metric space X, that . Now, by the triangle inequality for the interleaving distance, by the inequality above, and by the Gromov-Hausdorff stability of X H k • VR(X, •) [CdSO14, CCSG + 09], we have Since this holds for any δ > 0, we find d I (H k • VR fin ∞ (X; •), H k • VR(X, •)) = 0. This implies that the bottleneck distance between persistence diagrams is 0, and the (undecorated) diagrams are in fact equal (see for instance [CdSGO16,Theorem 4.20]). We can apply Corollary 5.3 to get the same result for VR ∞ (X; •) (measures with infinite support), and the same proof works in theČech case. We state this as the following theorem.
Corollary 5.10. For any totally bounded metric space X and any integer k ≥ 0, we have Remark 5.11. Whereas the ∞-metric thickenings VR ∞ (X; •) and the simplicial complexes VR(X, •) yield persistence modules with an interleaving distance of 0, we are in the interesting landscape where the metric thickenings VR p (X; •) may yield something new and different for p < ∞. For example, in Section 7 we explore the new persistence modules that arise for p-metric thickenings of Euclidean spheres (S n , 2 ) in the case p = 2.
Question 5.14. [LMO20, Theorem 5.2] states that for X compact, bars in the persistent homology of the simplicial complex filtration VR(X, •) are of the form (a, b] or (a, ∞). Is the same true for VR ∞ (X; •)? Note that we are using the < convention, i.e. a simplex is included in VR(X; r) when its diameter is strictly less than r, and a measure is included in VR ∞ (X; r) when the diameter of its support is strictly less than r.

The proof of stability
As mentioned above, the construction of interleaving maps between filtrations of metric thickenings is more intricate than in the case of simplicial complexes. A main idea behind the proof of our stability result, Theorem A, is to approximate the metric space X by a finite subspace (a net) U . The advantage is that it is easier to construct a continuous map with domain P U than one with domain P X . The crucial next step is the construction of a continuous map from P X to P U by using a partition of unity subordinate to an open covering by δ-balls. This map distorts distances by a controlled amount, allowing us to approximate measures of P X in P U .
We begin with some necessary lemmas. We have the continuous map Φ U : P X P U defined by For any q ∈ [1, ∞], we have d X W,q (α, Φ U (α)) < δ.
For the continuity, since the weak topology on P X and P U can be metrized, it suffices to show for a weakly convergent sequence α n ∈ P X , that the image Φ U (α n ) is also weakly convergent. As U is a finite metric space, it suffices to show that for any fixed u 0 ∈ U , the sequence of real numbers Φ U (α n )({u 0 }) n is itself convergent. Note that Since ζ U u 0 is a bounded continuous function on X, we obtain the desired convergence through the weak convergence of α n .
Lastly, we must show that d X W,q (α, Φ U (α)) < δ. For any u ∈ U , we use the notation w α u := X ζ U u (x)α(dx), so that Φ U (α) = u∈U w α u · δ u and From this we get, The first inequality is by Lemma 4.2. The last one comes from the fact that each 1 w α u ζ U u α is a probability measure supported in B(u; δ), and thus d X For any two totally bounded metric spaces X and Y , through partitions of unity we can build continuous maps between P X and P Y , as follows. Let X and Y be two totally bounded metric spaces, let η > 2 d GH (X, Y ), and let δ > 0. We fix finite δ-nets U ⊆ X of X and V ⊆ Y of Y . By the triangle inequality, we have d GH (U, V ) < η 2 + 2δ. So there exist maps ϕ : U V and ψ : V U with max dis(ϕ), dis(ψ), codis(ϕ, ψ) ≤ η + 4δ.
We then define the maps Φ : P X P Y and Ψ : P Y P X to be where ι U : P U P X and ι V : P V P Y are inclusions and Φ U and Φ V are the maps defined in Lemma 5.15: Then both Φ and Ψ are continuous maps by Lemma 5.15 and by the continuity of pushforwards of continuous maps.
The use of the finite δ-nets U and V is important because they can be compared by continuous maps ϕ and ψ, which yield continuous maps ϕ and ψ . The following lemma shows that Φ and Ψ are homotopy equivalences.
Lemma 5.16. With the above notation, we have homotopy equivalences Ψ • Φ id P X and Proof. The homotopies H X t and H Y t are continuous by Proposition 2.4, since Φ and Ψ are continuous. We will only present the estimate d X W,q (H X t (α), α) < η + 6δ, as the other inequality can be proved in a similar way. We first calculate the expression for Ψ • Φ(α), obtaining where we again use the notation w α u := X ζ U u (x)α(dx). We have Now we show that H X t (α) is a convex combination of probability measures in P X and hence itself lies in P X . That is, the sum of coefficients in front of all δ ψ(v) and α in the above formula for H X t is 1: Next, we will bound the q-Wasserstein distance d X W,q (H X t (α), α). For this, we first rewrite α as follows: Based on these observations, we then apply the inequality from Lemma 4.2 to obtain For each non-zero term in the summand of the last expression, we have d Y (v, ϕ(u)) < δ which implies d X (ψ(v), u) < 5δ + η via the codistortion assumption. Therefore, d X W,q δ ψ(v) , δ u < 5δ + η as well. Moreover, since 1 w α u ζ U u α is a probability measure supported in the δ-ball centered at the point u, we have d X W,q δ u , 1 w α u ζ U u α < δ. Therefore, we have d X W,q (H X t (α), α) < 6δ + η.
Remark 5.17. The above lemma still holds if we replace P X by P fin X and P Y by P fin Y as all related maps naturally restrict to the set of finitely supported measures, P fin X .
We are now ready to prove our main result, Theorem A.
Proof of Theorem A. For i an L-controlled invariant, for totally bounded metric spaces X and Y , and for any integer k ≥ 0, we must show . We prove only the first line above, as the finitely supported case in the second line has a nearly identical proof. The inequality involving d I and d HT follows from Lemma 2.9. Hence it suffices to prove the inequality d HT (P X , i X ), Following the construction in Lemma 5.16, let η > 2 · d GH (X, Y ) and δ > 0. We fix finite δ-nets U ⊂ X of X and V ⊂ Y of Y . By the triangle inequality, we have d GH (U, V ) < η 2 + 2δ, and so there exist maps ϕ : U V and ψ : V U with max dis(ϕ), dis(ψ), codis(ϕ, ψ) ≤ η + 4δ.
For any δ > 0, we will show that (P X , i X ) and (P Y , i Y ) are (η + 6δ) L-homotopy equivalent. By the definition of a δ-homotopy (Definition 2.6), it suffices to show that • the map Ψ • Φ : P X P X is (2η + 12δ) L-homotopic to id P X with respect to (i X , i X ), • the map Φ • Ψ : P X P X is (2η + 12δ) L-homotopic to id P Y with respect to (i Y , i Y ).
We will only present the proof for the first and third items; the other two can be proved similarly. We have where the first inequality is from the stability of the invariant i under pushforward and from the bound on the distortion of ϕ, and where the second inequality is from the stability of the invariant with respect to Wasserstein distance and from the bound d X W,q (α, Φ U (α)) < δ in Lemma 5.15. This proves the first item.
For the third item, we use the homotopy H X t in Lemma 5.16. Then it suffices to show for any fixed t ∈ [0, 1], the map H X t is a (2η + 12δ) L-map from (P X , i X ) to (P X , i X ), that is, i X (H X t (α)) ≤ i X (α) + (2η + 12δ) L. This comes from the inequality d X W,∞ (H X t (α), α) < η + 6δ from Lemma 5.16, and the from the stability assumption of i with respect to Wasserstein-distance. Now, since (P X , i X ) and (P Y , i Y ) are (η + 6δ) L-homotopy equivalent for any η > 2 · d GH (X, Y ) and δ > 0, it follows from the definition of the d HT distance (Definition 2.7) that d HT (P X , i X ), (P Y , i Y ) ≤ 2L · d GH (X, Y ). This completes the proof.

A Hausmann type theorem for 2-Vietoris-Rips and 2-Čech thickenings of Euclidean submanifolds
Hausmann's theorem [Hau95] states that if X is a Riemannian manifold and if the scale r > 0 is sufficiently small (depending on the curvature X), then the Vietoris-Rips simplicial complex VR(X; r) is homotopy equivalent to X. A short proof using an advanced version of the nerve lemma is given in [Vir21b]. Versions of Hausmann's theorem have been proven for ∞-metric thickenings in [AAF18], where X is equipped with the Riemannian metric, and in [AM19], where X is equipped with a Euclidean metric. In this section we prove a Hausmann type theorem for the 2-Vietoris-Rips and 2-Čech metric thickenings. Our result is closest to that in [AM19] (now with p = 2 instead of p = ∞): we work with any Euclidean subset X of positive reach. This includes (for example) any embedded C k submanifold of R n for k ≥ 2, with or without boundary [Thä08]. We begin with a definition and some lemmas that will be needed in the proof. Let α be a measure in P 1 (R n ), the collection of all Radon measures with finite first moment. For any coordinate function Therefore, the Euclidean mean map m : P 1 (R n ) R n given by the following vector-valued integral is well-defined: Lemma 6.1. Let X be a metric space and let f : X R n be a bounded continuous function. Then the induced map m • f : P X R n is continuous. Proof. For a sequence α n that weakly converges to α, we have the following vector-valued integral: As f is bounded and continuous, the above limit converges to m • f (α) as α n converges to α. Therefore, the map m • f is continuous.
Lemma 6.2. Let α be a probability measure in P 1 (R n ). Then for p ∈ [1, ∞], there is some We consider the following formula: So there must be some z ∈ supp(α) with m(α) − z ≤ diam 1 (α).
Lemma 6.3. For any probability measure α ∈ P 1 (R n ) and for any x ∈ R n , we can write the associated squared 2-Fréchet function F 2 α,2 (x) as Proof. By the linearity of the inner product, we have Recall the medial axis of X is defined as the closure The reach τ of X is the closest distance τ := inf x∈X,y∈med(X) x − y between points in X and med(X). Let U τ (X) be the τ -neighborhood of X in R n , that is, the set of points y ∈ R n such that there is some x ∈ X with y − x < τ . The definition of reach implies that for any point y in the open neighborhood U τ (X), there is a unique closest point x ∈ X. The associated nearest projection map π : U τ (X) X is continuous; see [Fed59, Theorem 4.8(8)] or [AM19, Lemma 3.7].
Lemma 6.4. Let X be a bounded subset of R n with reach τ (X) > 0. Let α ∈ P X have its Euclidean mean m(α) in the neighborhood U τ (X). Then along the linear interpolation family where φ is the composition π • m : P X X, both diam 2 and rad 2 obtain their maximum at t = 0.
Proof of Lemma 7.1. We can calculate diam 2 (α) as follows: The calculation for rad 2 (α) gives We now see that for spheres with euclidean metric, both the 2-Vietoris-Rips and the 2-Čech metric thickenings attain only two homotopy types: Theorem D. For any r ≤ √ 2, the isometric embeddings S n VR 2 ((S n , 2 ); r) and S n Č 2 ((S n , 2 ); r) are homotopy equivalences. When r > √ 2, the spaces VR 2 ((S n , 2 ); r) anď C 2 ((S n , 2 ); r) are contractible. By restricting to finitely supported measures, we get the same result for VR fin 2 ((S n , 2 ); r) andČ fin 2 ((S n , 2 ); r). Remark 7.2. A similar phenomenon is found for the Vietoris-Rips filtration of (S n , ∞ ) where VR((S n , ∞ ); r) has the homotopy type of S n when r ≤ 2 √ n+1 and becomes contractible when r > 2 √ n+1 ; see [LMO20, Section 7.3]. In contrast, the homotopy types of the Vietoris-Rips andČech filtrations of geodesic circle include all possible odd-dimensional spheres [AA17]. The first new homotopy type of the ∞-Vietoris-Rips metric thickening of the n-sphere (with either the geodesic or the Euclidean metric) is known [AAF18, Theorem 5.4], but so far only for a single (non-persistent) scale parameter, and only when using the convention diam ∞ (α) ≤ r instead of diam ∞ (α) < r.
We note that Theorem C only implies that VR 2 ((S n , 2 ); r) S n for r ≤ τ (S n ) = 1, whereas Theorem D extends this range to r ≤ √ 2.
Remark 7.3. The above calculation gives, for any two positive integers n = m, that Remark 7.4. In a more detailed analysis, Proposition 9.13 of [LMS21] provides the lower bound 1 2 for d GH ((S n , 2 ), (S m , 2 )) (when n = m), which is larger than the lower bound √ 2 4 given in (7). As for the geodesic distance case, one can use [LMS21, Corollary 9.8 (1)] to obtain d GH (S n , S m ) ≥ arcsin( √ 2/4) from (7), where d GH (S n , S m ) is the Gromov-Hausdorff distance between spheres endowed with their geodesic distances for any 0 < m < n. A better lower bound is found in [LMS21]: which arises from considerations other than stability. The lower bound given by the quantity in the middle is shown to coincide with the exact Gromov-Hausdorff distance between S 1 and S 2 , between S 1 and S 3 , and also between S 2 and S 3 .

Bounding barcode length via spread
The spread of a metric space is defined by Katz [Kat83], and used in [LMO20, Section 9] to bound the length of intervals in Vietoris-Rips simplicial complex persistence diagrams. In this section we identify a measure theoretic version of the notion of spread, and we use it to bound the length of intervals in p-Vietoris-Rips and p-Čech metric thickening persistence diagrams.
Definition 8.1. The p-spread of a bounded metric space X is defined as Note that for any 1 ≤ p ≤ q ≤ ∞, we have Remark 8.2. When p ∈ [1, ∞), the space P fin X is dense in P p,X , the set of Radon measures with finite moment of order p (see [Bog18,Corollary 3.3.5]). Therefore, for a bounded metric space, the p-spread is just the radius of the metric space (P X , d X W,p ).
Proposition 8.3. Let X be a bounded metric space and let i be an L-controlled invariant. For any r > 0 and any ξ > spread ∞ (X), the space [P X , i X ; r] can be contracted inside of [P X , i X ; r + 2Lξ]. In particular, any homology class of [P X , i X ; r] will vanish in [P X , i X ; r + 2Lξ].
Proof. As ξ > spread ∞ (X), there is some α ξ ∈ P fin X such that for all α ∈ P X we have d X W,∞ (α ξ , α) < ξ. Consider the linear homotopy This gives a homotopy between the inclusion from [P X , i X ; r] to [P X , i X ; r + 2Lξ] and a constant map, and therefore implies our conclusion so long as the homotopy is well-defined. It then suffices to show i X (h t (α)) < r + 2Lξ. This comes from the stability condition of the invariant with respect to Wasserstein distances and from Lemma 4.2: Remark 8.4. It is not difficult to see that for the i q,p invariants we can improve the bound ξ > spread ∞ (X) to ξ > max{spread p (X), spread q (X)}.
We also have the following stronger contractibility conclusion for VR p (X; •) andČ p (X; •).
Theorem E. Let X be a bounded metric space. For any p ∈ [1, ∞] and any r > spread p (X), both VR p (X; r) andČ p (X; r) are contractible.
Let i be an L-controlled invariant and let max(i X ) be the maximum of the function i X on P X , for X a bounded metric space. For any r > max(i X ), we have [P X , i X ; r] = P X , and therefore [P X , i X ; r] is contractible. Inspired by the definition of the spread, we have the following result.
Lemma 8.6. Let X be a metric space, let i be an L-controlled invariant, and let U be a finite subset of X. Then for any ξ > max d H (U, X), max(i U )−r 2L , the space [P X , i X ; r] is contractible in [P X , i X ; r + 2Lξ]. In particular, any homology class of [P X , i X ; r] will vanish in [P X , i X ; r + 2Lξ].
Proof. As ξ > d H (U, X), the balls {B(u; ξ)} u∈U form a open covering of X. We choose a partition of unity subordinate to the covering and build the map Φ U : [P X , i X ; r] P U as in Lemma 5.15. Since r + 2L ξ > max(i U ), we know P U = [P U , i U ; r + 2Lξ]. Since the inclusion map ι U : U X is of zero distortion, we have i X ((ι U ) (β)) ≤ i U (β) ≤ max(i U ) for all β ∈ P U . This implies that the contractible set P U is inside [P X , i X ; r + 2Lξ], and so we have the following diagram.
is homotopy equivalent to a constant map. To obtain this conclusion, it thus suffices to show that ι U • Φ U is homotopy equivalent to the inclusion given by the structure map ν X r,r+2L ξ . Let's consider the linear homotopy From the stability property of i, Lemma 4.2, and the estimate in Lemma 5.15, we have This shows that the homotopy h t from ι U • Φ U to ν X r,r+2L ξ is well-defined. Remark 8.7. For the 1-controlled i q,p invariant on a bounded metric space X, the maximum of i X q,p is bounded by diam(X). Therefore, for any finite subset U we have Hence, the previous lemma implies that the lifetime of features in [P X , i X q,p ; •] is bounded by 2 spread(X). In the diam p case, the factor 2 can be removed, as we show next, and therefore matches the result in [LMO20, Proposition 9.6] for Vietoris-Rips simplicial complexes.
Proof. As ξ > spread(X), there exists some U ⊂ X such that diam(U ) < ξ and d H (U, X) < ξ. As the maximum of diam U p on P U is bounded by diam(U ), the contractible subset P U lies inside VR p (X; r + ξ), and therefore ι U • Φ U is homotopy equivalent to a constant map. Let h t be the linear homotopy used in Lemma 8.6. What is left to get a homotopy equivalence between ι U • Φ U and ν X r,r+ξ is to show diam p (h t (α)) < r + ξ. This comes from the following calculation: The following result shows the lifetime of features ofČ p (X; •) is also bounded by the metric spread of X, spread(X).
Proof. As ξ > spread(X), there exists some U ⊂ X such that diam(U ) < ξ and d H (U, X) < ξ. The maximum of rad U p on P U is bounded by diam(U ), and the contractible subset P U lies insideČ p (X; r + ξ). This shows ι U • Φ U is homotopy equivalent to a constant map. Let h t = (1 − t) α + t Φ U (α) be the linear homotopy used in Lemma 8.6. What is left is to show is rad p (h t (α)) < r + ξ. By Lemma 4.9 and Lemma 4.2, we have

Conclusion
Filtrations, i.e. increasing sequences of spaces, play a foundational role in applied and computational topology, as they are the input to persistent homology. To produce a filtration from a metric space X, one often considers a Vietoris-Rips orČech simplicial complex, with X as its vertex set, as the scale parameter increases. Since a point in the geometric realization of a simplicial complex is a convex combination of the vertices of the simplex [x 0 , x 1 , . . . , x k ] in which it lies, each such point can alternatively be identified with a probability measure: a convex combination of Dirac delta masses δ x 0 , δ x 1 , . . . , δ x k . We can therefore re-interpret the Vietoris-Rips andČech simplicial complex filtrations instead as filtrations in the space of probability measures, which are referred to as the Vietoris-Rips andČech metric thickenings. In [AAF18] it is argued that the metric thickenings have nicer properties for some purposes: for example, the inclusion from metric space X into the metric thickening is always an isometry onto its image, whereas an inclusion from metric space X into the vertex set of a simplicial complex is not even continuous unless X is discrete. In this paper, we prove that these two perspectives are compatible: the ∞-Vietoris-Rips (resp.Čech) metric thickening filtration has the same persistent homology as the Vietoris-Rips (resp.Čech) simplicial complex filtration when X is totally bounded. Therefore, when analyzing these filtrations, one can choose to apply either simplicial techniques (simplicial homology, simplicial collapses, discrete Morse theory) or measure-theoretic techniques (optimal transport, Karcher or Fréchet means), whichever is more convenient for the task at hand.
The measure-theoretic perspective motivates new filtrations to build on top of a metric space X. Though the Vietoris-Rips simplicial complex filtration is closely related (at interleaving distance zero) to the metric thickening filtration obtained by looking at sublevelsets in the space of probability measures of the ∞-diameter functional, one can instead consider sublevelsets of the p-diameter functional for any 1 ≤ p ≤ ∞. The same is true upon replacing Vietoris-Rips withČech and replacing p-diameter with p-radius. These relaxed p-Vietoris-Rips and p-Čech metric thickenings enjoy the same stability results underlying the use of persistent homology: nearby metric spaces produce nearby persistence modules. The generalization to p < ∞ is a useful one: though determining the homotopy types of ∞-Vietoris-Rips thickenings of n-spheres is a hard open problem, we give a complete description of the homotopy types of 2-Vietoris-Rips thickenings of n-spheres for all n. We also prove a Hausmann-type theorem in the case p = 2, and ask if the p < ∞ metric thickenings may be amenable to study using tools from Morse theory.
More generally, one can consider sublevelsets of any L-controlled function on the space of probability measures on X. We prove stability in this much more general context. This allows one to consider metric thickenings that are tuned to a particular task, perhaps incorporating other geometric notions besides just proximity, such as curvature, centrality, eccentricity, etc. One can design an L-controlled functional to highlight specific features that may be useful for a particular data science task.
We hope these contributions inspire more work on metric thickenings and their relaxations. We end with some open questions.
1. For X totally bounded, is the p = ∞ metric thickening VR ∞ (X; r) homotopy equivalent to the simplicial complex VR(X; r), and similarly isČ ∞ (X; r) homotopy equivalent toČ(X; r)? Note we are using the < convention.
2. For X totally bounded, is VR p (X; r) homotopy equivalent to VR fin p (X; r), isČ p (X; r) homotopy equivalent toČ fin p (X; r), and for i a controlled invariant is [P X , i X ; r] homotopy equivalent to [P fin X , i X ; r]?
3. Is there an analogue to the Hausmann-type Theorem C which holds for p ∈ (2, ∞)?
The case p = ∞ was tackled in [AM19]. In a similar spirit, it seems interesting to explore whether analogous theorems hold when the ambient space is a more general Hadamard space instead of R d . 9. For X finite with n + 1 points, the space P X is an n-simplex in R n+1 where coordinates are the weights of a measure at each point. In this case, diam p is a quadratic polynomial on R n+1 and rad p is the minimum of n + 1 linear equations. Therefore, both VR p (X; r) andČ p (X; r) are semi-algebraic sets in R n+1 . Can one use linear programming along with the results of Appendix A.3 to calculate the homology groups ofČ(X; r), and the work on quadratic semi-algebraic sets [BPR08, Bas08, BCL18] to calculate the homology groups of VR p (X; r)? See also the recent paper [BK22], which provides a singly exponential complexity algorithm for computing the sub-level set persistent homology induced by a polynomial on a semi-algebraic set up to some fixed homological dimension.
Definition A.1. Let X be a metric space. The Lévy-Prokhorov metric on P X is given by the formula Here is the open ε-neighborhood of E in P X , and B(X) is the Borel σ-algebra.
We state two theorems that we will use 3 Theorem ([Bog18, Theorem 3.1.4]). The weak topology on the set P X is generated by the Lévy-Prokhorov metric. 4 Theorem ([GS02, Theorem 2]). On a metric space X, for any α, β in P X , one has Corollary A.2. On a bounded metric space X, for any q ∈ [1, ∞), the q-Wasserstein metric generates the weak topology on P X .
Proof. From [GS02, Theorem 2] and [Bog18, Theorem 3.1.4], we know d X W,1 generates the weak topology. For other values of q, notice that for any coupling µ between α and β, we have Along with d X W,1 ≤ d X W,q , this implies that on a bounded metric space, all q-Wasserstein metrics with q finite are equivalent and generate the weak topology on P X .

A.2 Min-type Morse theory
The paper [BBK14] by Baryshnikov, Bubenik, and Kahle studies a Morse theory for mintype functions; see also [GR97,Bry78,Mat82]. The following notation is from [BBK14, Section 3.1]. Let X be a compact metric space (called the parameter space), let M be a compact smooth manifold perhaps with boundary, and let f : Let ∇f x : M R be the gradient of f x with respect to m. We furthermore assume that the function X × M R defined by (x, m) ∇f x (m) is continuous. There is then an analogue of Morse theory, called "min-type Morse theory", for the min-type function τ : M R defined by τ (m) = min x∈X f x (m).
Whereas [BBK14] uses min-type Morse theory to study configurations of hard spheres, we instead propose the use of min-type Morse theory to study p-Čech metric thickenings, as follows. Let X be a finite metric space with n + 1 points. Define M = ∆ n to be the n-simplex on n + 1 vertices; a point m ∈ ∆ n is given in barycentric coordinates as m = (m 0 , . . . , m n ) with m i ≥ 0 and i m i = 1. For x ∈ X, let f x : ∆ n R be defined by f x (m) = i m i d p X (x, x i ); note this is equal to the p-th power of the p-Frechét function, . Note that each gradient ∇f x : ∆ n R is linear and hence continuous, and so the joint function X × ∆ n R given by (x, m) ∇f x (m) is continuous since X is discrete. The function τ : ∆ n R is then defined by τ (m) = min x∈X f x (m); note that this is equal to rad p p (α) = inf x∈X F p α,p (x) for α = i m i δ x i . By Lemma 4.11 and its proof, we have not only a homeomorphism P X ∼ = ∆ n , but also a homeomorphism meaning that the p-Čech metric thickenings are homeomorphic to the sublevel sets of the min-type function τ . In this setting, we have the additional convenience that each function f x is affine.
Question A.3. Can the machinery from [BBK14] be used to prove new results about p-Čech metric thickenings, such as homotopy types? A first step in this direction might be to use their balanced criterion (which derives from Farkas' lemma) to help identify which points are topological regular points or critical points of τ .
Question A.4. We have restricted to X finite (n + 1 points) so that M = ∆ n will be a manifold with boundary. Can one build up towards letting X be a manifold, such as a circle or n-sphere?
A.3 Finite p-Čech metric thickenings are homotopy equivalent to simplicial complexes The p-Vietoris-Rips and p-Čech matric thickenings considered in this paper are based on the corresponding simplicial complexes and are closely related to them. The most direct relationship is given by Lemma 4.12 in the case of finite metric spaces and p = ∞. Further similarities in the case of totally bounded metric spaces and p = ∞ are observed in the persistence diagrams, as shown in Corollary 5.10. For p < ∞, the metric thickenings are not as directly related to the corresponding simplicial complexes. However, in this appendix, we establish that all p-Čech metric thickenings on finite metric spaces are homotopy equivalent to simplicial complexes (although generally not the correspondingČech simplicial complexes).
Theorem F. Let (X, d X ) be a finite metric space with n + 1 points. For any p ∈ [1, ∞] and any r > 0,Č p (X; r) is homotopy equivalent to a simplicial complex on n + 1 vertices, consisting of the simplices contained in the homeomorphic image ofČ p (X; r) in the standard n-simplex.
Proof. The result holds for the case p = ∞ by Lemma 4.12, so we will suppose p ∈ [1, ∞). We begin with some background notation and observations. Let σ be a simplex in a Euclidean space, let z 0 ∈ σ, and let σ z 0 be the union of the closed faces of σ not containing z 0 (we always have σ z 0 ⊆ ∂σ, and σ z 0 = ∂σ if z 0 is in the interior of σ). Then there is a continuous function P : σ \ {z 0 } σ z 0 defined by projecting radially from z 0 . Furthermore, if C ⊆ σ is convex and contains z 0 , then for any z ∈ σ \ C, the line segment connecting z and P (z) is contained in σ \ C, since this lies in the line segment connecting z 0 and P (z). Therefore a linear homotopy shows that P | σ\C : σ \ C σ z 0 \ C is a strong deformation retraction. We will apply such retractions successively to a simplex and its faces as described below.
Let X = {x 0 , . . . , x n }. If α = i a i δ x i , then using the notation for the Fréchet function from Section 2, we have F p α,p (x j ) = i a i d p X (x i , x j ). Thus, α ∈Č p (X; r) if and only if i a i d p X (x i , x j ) < r p for some j. Let ∆ = {(y 0 , . . . , y n ) ∈ R n+1 | i y i = 1, y i ≥ 0 for all i} be the standard n-simplex in R n+1 . By Lemma 4.11,Č p (X; r) is homeomorphic to Equivalently, Y is a sublevel set of the function ρ : ∆ R given by ρ(y 0 , . . . , y n ) = min j i We note that Y contains the vertices of ∆ by the assumption that r > 0. If Y = ∆, theň C p (X; r) ∼ = ∆, and ∆ is homeomorphic to a simplex with vertex set X, as required. If not, then ∆ \ Y is convex as it is the intersection of half-spaces and ∆, so we may project radially from any point z 0 ∈ ∆ \ Y as above. This shows Y Y ∩ ∆ z 0 , where ∆ z 0 is the union of the closed faces of ∆ not containing z 0 .
Since Y ∩ ∆ z 0 is contained in the boundary of ∆, we will next verify that we can define retractions within the (n − 1)-dimensional faces of ∆ contained in ∆ z 0 that contain a point not in Y . More generally, we will repeat for successively lower dimensional faces, inductively obtaining a sequence of strong deformation retractions Y n Y n−1 · · · Y 0 , where Y n = Y . Each Y k will consist of all closed faces of ∆ contained in Y , along with some subset of the remaining k-dimensional closed faces intersected with Y . Thus, Y 0 will be a simplicial complex consisting of the closed faces of ∆ contained Y and will be homotopy equivalent tǒ C p (X; r), as required.
We will use induction, so suppose Y k meets the description above. Let σ 1 , . . . , σ m be those remaining k-dimensional faces of ∆ whose intersections with Y are contained in Y k and that contain a point in their interior that is not in Y k ; let τ 1 , . . . , τ m be the remaining k-dimensional faces whose intersections with Y are contained in Y k and that have only some boundary point not in Y k . On each σ l , we may choose an interior point not in Y k and project Figure 4: The sequence of deformation retractions used in the proof of Theorem F collapses a subset of a simplex to a simplicial complex on its vertices. In this example, n = 2 and the sequence of deformation retractions is Y 2 Y 1 Y 0 .
radially from this point as above. The homotopy is constant on the boundary, so we may define a homotopy on Y k that simultaneously retracts each σ l ∩ Y k to ∂σ l ∩ Y k and is constant on all points not in any σ l . Thus, Y k deformation retracts onto a subset Y k consisting of faces of ∆ that are contained in Y , along with τ l ∩ Y k for all l.
Next, choose any point y ∈ ∂τ 1 \ Y k . Note that projecting radially from y as above affects any simplex in ∂τ 1 that contains y, so we must extend this to a homotopy H on Y k in a way that is consistent on the simplices containing y. For any τ l containing y, let H be the defined on τ l ∩ Y k by the radial projection from y, as above. For any two τ l and τ l containing y, the two definitions of H on τ l ∩ τ l ∩ Y are consistent, as they are both linear homotopies. We also let H be constant on any point in a face of ∆ not containing y: this includes all faces of ∆ contained in Y and all τ l ∩ Y such that y / ∈ τ l . The definitions are consistent on their overlap, since if τ l contains y, then radial projection from y is constant on the faces of τ l that do not contain y. Therefore H is a well-defined homotopy, which shows that Y k deformation retracts onto the subset that excludes the interiors of all τ l containing y. We can repeat for the remaining set of τ l until all have been retracted. Composing these deformation retractions, we have shown that Y k deformation retracts onto a subset Y k−1 , consisting of faces of ∆ contained in Y along with the intersections of Y with a subset of the remaining (k − 1)-dimensional faces of ∆ (subsets of the boundaries of those k-dimensional simplices that were collapsed). This completes the inductive step, so we obtain the sequence Y n Y n−1 · · · Y 0 of strong deformation retractions as required.
Č p (X; r 1 )Č p (X; r 2 ) S(X; r 1 ) S(X; r 2 ) This shows that the induced maps on homology commute, giving a morphism of persistence modules. Since each vertical map is an isomorphism, this is an isomorphism of persistence modules.
These results show that p-Čech persistent homology can be computed, at least in principle. The complex S(X; r) in Corollary A.5 consists of all simplices contained in the r p -sublevel set of the function ρ in the proof of Theorem F. So finding the filtration S(X; •) would require finding the maximum of ρ on each face. Calculating persistent homology would involve finding the maximum of ρ on each face of the necessary dimensions.
A.4 Persistence diagrams of VR p (Z n+1 ; •) andČ p (Z n+1 ; •) In this section, we will calculate the persistence diagrams of the p-Vietoris-Rips and p-Čech metric thickenings of the metric space Z n+1 , consisting of n + 1 points with all interpoint distances equal to one. Let ∆ n be the n-dimensional simplex on n + 1 points, and let ∆ (k) n denote its k-skeleton.
Proof. We will use the following observation, for which we omit the proof. Let α be a measure in P Z m+1 , where m + 1 is any positive integer. Then both diam p and rad p will obtain the maximum m m+1 1 p only at the uniform measure supported on the m + 1 points of the metric space Z m+1 , i.e. the barycenter of the m-simplex P Z m+1 .
Let k be an integer where 0 ≤ k ≤ n − 1. For any r in k k+1 1 p , k+1 k+2 1 p , we will show both VR p (Z n+1 ; r) andČ p (Z n+1 ; r) can be deformation retracted to the k-skeleton of the n-simplex P Z n+1 , denoted ∆ (k) n . Since ( k k+1 ) 1 p < r ≤ ( k+1 k+2 ) 1 p , we know both VR p (Z n+1 ; r) andČ p (Z n+1 ; r) contain the k-skeleton of ∆ n , but not any higher-dimensional skeleta. We can then use the radial projection as in Corollary A.5 to work out the retraction. From the above observation, the barycenter of any simplex in ∆ n with dimension higher than k is not in VR p (Z n+1 ; r) andČ p (Z n+1 ; r). We can use these barycenters as the base point for radial projection. We start from ∆ n : if the radial projection based at its barycenter can be restricted to VR p (Z n+1 ; r) orČ p (Z n+1 ; r), then it will retract them onto their intersections with the (n−1)-skeleton (∆ n ) (n−1) . From the proof of Corollary A.5, we know the restriction is well-defined forČ p (Z n+1 ; r). Here, it is also well-defined for VR p (Z n+1 ; r), because our base point for the radial projection is an interior maximum of the quadratic function diam p p , and diam p p will be concave along any line that passes through the base point. This in turn shows the line segment connecting two points outside of VR p (Z n+1 ; r) is also outside of VR p (Z n+1 ; r). We can continue the retraction process inductively on VR p (Z n+1 ; r) ∩ ∆ (l) n andČ p (Z n+1 ; r) ∩ ∆ (l) n , where k < l ≤ n − 1, with radial projections based at barycenters of l-simplices of ∆ n . This results in a retraction onto VR p (Z n+1 ; r) ∩ ∆ (l−1) n andČ p (Z n+1 ; r) ∩ ∆ (l−1) n , respectively. Eventually, we will deformation retract onto VR p (Z n+1 ; r) ∩ ∆ Corollary A.7. Let n be a positive integer and let Z n+1 be the metric space consisting of n + 1 points with interpoint distance equal to 1. The persistence diagrams of VR p (Z n+1 ; •) andČ p (Z n+1 ; •) are the same and of the following form: The superscripts denote the multiplicity of a point in the persistence diagram.
Proof. For any integer k with 0 < k < n − 1, we know the homology of the k-skeleton of an n-simplex ∆ n is given by We get the result by combining with the previous result on the homotopy types of VR p (Z n+1 ; •) andČ p (Z n+1 ; •).
A.5 Zero-dimensional persistent homology of VR p (X; •) andČ p (X; •) For a finite metric space X, we will show that the 0-dimensional persistent homology of VR p (X; •) andČ p (X; •) are the same, and that they both recover the single-linkage clustering up to a constant factor related to p.
Lemma A.8. For X a finite metric space, the birth time for all intervals in the 0-dimensional barcodes of VR p (X; r) andČ p (X; r) is zero.
Proof. Since all delta measures δ x have diam p and rad p equal zero, it suffices to show for any measure α in VR p (X; r) (orČ p (X; r)), there is a path in VR p (X; r) (orČ p (X; r)) that connects α with some delta measure. For α a measure in VR p (X; r), we have A.6 Crushings and the homotopy type distance A crushing is a particular type of deformation retraction that doesn't increase distances. In this appendix, we consider the effects of a crushing applied to the metric space underlying a metric thickening. Let X be a metric space and A ⊆ X is a subspace. Following [Hau95], a crushing from X to A is defined as a distance non-increasing strong deformation retraction from X to A, that is, a continuous map H : X × [0, 1] X satisfying 1. H(x, 1) = x, H(x, 0) ∈ A, H(a, t) = a if a ∈ A.
When this happens we say that X can be crushed onto A.
In [Hau95, Proposition 2.2], Haussman proves that if X can be crushed onto A, then inclusion VR(A; r) VR(X; r) of Vietoris-Rips simplicial complexes is a homotopy equivalence. A similar result is proven for Vietoris-Rips andČech metric thickenings with p = ∞ in [AAF18, Appendix B]. We study how more general metric thickenings behave with respect to crushings. We begin with some preliminaries.
Lemma A.10. Let X be a complete separable metric space and let φ be an L-Lipschitz function that is absolutely bounded by a constant C > 0. Then for any α, β ∈ P X , we have Proof. Let ε = d LP (α, β), then by the expression of the Lévy-Prokhorov metric given in [Bog18, Theorem 3.1.5]), there is a coupling between α and β by µ ∈ P X×X such that µ({(x, x ) ∈ X × X | d X (x, x ) > ε}) ≤ ε.
Let E be the set {(x, x ) ∈ X × X | d X (x, x ) > ε}. Then we have Proposition A.11. Let X be a complete separable metric space such that there is a crushing from X onto a subset A ⊂ X. Then there is an induced deformation retraction from P X onto P A .
Proof. Let H be a crushing from X to A, so H is a continuous map from X × [0, 1] to X. We use the notation f t (x) to denote the map H(x, t) : X X for any fixed t in [0, 1]. Then we can define a mapH : P X × [0, 1] P X viã H(α, t) = (f t ) (α).
For continuity, let (α n , t n ) be a sequence that converges to (α ∞ , t ∞ ). Then for any bounded continuous function γ(x) on X, we have Without loss of generality, we may assume γ is 1-Lipschitz and absolutely bounded by C > 0. Therefore, every γ • f tn (x) is 1-Lipschitz and bounded by C for any n. We use the notation I i,j to denote X γ • f t i (x) α j (dx). Then |I i,j | is uniformly bounded by C. Lemma A.10 then shows that for any i, we have The uniform bound on |I i,∞ | implies I i,∞ converges to I ∞,∞ . For any ε > 0, we can find an N such that for any n > N , |I n,∞ − I ∞,∞ | ≤ ε 2 and d LP (α n , α ∞ ) ≤ ε 2(1 + 2C) .
In the same spirit of [AAF18, Lemma B.1], we can apply the above deformation retraction to the sublevel set filtrations of a set of invariants that includes i q,p and rad p .
Theorem G. Let i be an invariant such that for any metric spaces X and Y and any 1-Lipschitz map f : X Y , the induced map on P X does not increase the values of i. More precisely, for any α ∈ P X , we require i Y (f (α)) ≤ i X (α). Then for any complete separable metric space X and any subset A such that X can crushed onto A, we have d HT (P X , i X ), (P A , i A ) = 0.