#### Vol. 10, No. 1, 2017

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On an isoperimetric-isodiametric inequality

### Andrea Mondino and Emanuele Spadaro

Vol. 10 (2017), No. 1, 95–126
##### Abstract

The Euclidean mixed isoperimetric-isodiametric inequality states that the round ball maximizes the volume under constraint on the product between boundary area and radius. The goal of the paper is to investigate such mixed isoperimetric-isodiametric inequalities in Riemannian manifolds. We first prove that the same inequality, with the sharp Euclidean constants, holds on Cartan–Hadamard spaces as well as on minimal submanifolds of ${ℝ}^{n}$. The equality cases are also studied and completely characterized; in particular, the latter gives a new link with free-boundary minimal submanifolds in a Euclidean ball. We also consider the case of manifolds with nonnegative Ricci curvature and prove a new comparison result stating that metric balls in the manifold have product of boundary area and radius bounded by the Euclidean counterpart and equality holds if and only if the ball is actually Euclidean.

We then consider the problem of the existence of optimal shapes (i.e., regions minimizing the product of boundary area and radius under the constraint of having fixed enclosed volume), called here isoperimetric-isodiametric regions. While it is not difficult to show existence if the ambient manifold is compact, the situation changes dramatically if the manifold is not compact: indeed we give examples of spaces where there exists no isoperimetric-isodiametric region (e.g., minimal surfaces with planar ends and more generally ${C}^{0}$-locally asymptotic Euclidean Cartan–Hadamard manifolds), and we prove that on the other hand on ${C}^{0}$-locally asymptotic Euclidean manifolds with nonnegative Ricci curvature there exists an isoperimetric-isodiametric region for every positive volume (this class of spaces includes a large family of metrics playing a key role in general relativity and Ricci flow: the so-called Hawking gravitational instantons and the Bryant-type Ricci solitons).

Finally we prove the optimal regularity of the boundary of isoperimetric-isodiametric regions: in the part which does not touch a minimal enclosing ball, the boundary is a smooth hypersurface outside of a closed subset of Hausdorff codimension $8$, and in a neighborhood of the contact region, the boundary is a ${C}^{1,1}$ hypersurface with explicit estimates on the ${L}^{\infty }$ norm of the mean curvature.

##### Keywords
isoperimetric inequality, isodiametric inequality, Ricci curvature, regularity nonlinear elliptic PDE
##### Mathematical Subject Classification 2010
Primary: 49J40, 49Q10, 49Q20, 35J93