For a convex body
$K$ in a
Euclidean vector space
$\mathcal{X}$
of dimension
$n$
($\ge 2$),
we define two subarithmetic monotonic sequences
${\left\{{\sigma}_{K,k}\right\}}_{k\ge 1}$ and
${\left\{{\sigma}_{K,k}^{o}\right\}}_{k\ge 1}$ of functions on
the interior of
$K$.
The
$k$th
members are “mean Minkowski measures in dimension
$k$” which are
pointwise dual:
${\sigma}_{K,k}^{o}\left(z\right)={\sigma}_{{K}^{z},k}\left(z\right)$,
where
$z\in int\phantom{\rule{0.3em}{0ex}}K$, and
${K}^{z}$ is the dual (polar)
of
$K$ with respect to
$z$. They are measures
of (anti)symmetry of
$K$
in the following sense:
$$1\le {\sigma}_{K,k}\left(z\right),{\sigma}_{K,k}^{o}\left(z\right)\le \frac{k+1}{2}.$$
The lower bound is attained if and only if
$K$ has a
$k$dimensional
simplicial slice or simplicial projection. The upper bound is attained if and only if
$K$ is symmetric
with respect to
$z$.
In 1953 Klee showed that the lower bound
${\mathfrak{m}}_{K}^{\ast}>n1$ on the Minkowski measure
of
$K$ implies that there are
$n+1$ affine diameters meeting
at a critical point
${z}^{\ast}\in K$.
In 1963 Grünbaum conjectured the existence of such a point in the interior of any
convex body (without any conditions). While this conjecture remains open (and
difficult), as a byproduct of our study of the dual mean Minkowski measures, we
show that
$$\frac{n}{{\mathfrak{m}}_{K}^{\ast}+1}\le {\sigma}_{K,n1}^{o}\left({z}^{\ast}\right)$$
always holds, and for sharp inequality Grünbaum’s conjecture is valid.
