Vol. 292, No. 1, 2018

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Dual mean Minkowski measures and the Grünbaum conjecture for affine diameters

Qi Guo and Gabor Toth

Vol. 292 (2018), No. 1, 117–137
Abstract

For a convex body $K$ in a Euclidean vector space $\mathsc{X}$ 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 ${\mathfrak{m}}_{K}^{\ast }>n-1$ 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

$\frac{n}{{\mathfrak{m}}_{K}^{\ast }+1}\le {\sigma }_{K,n-1}^{o}\left({z}^{\ast }\right)$

always holds, and for sharp inequality Grünbaum’s conjecture is valid.

Keywords
convex body, dual, Minkowski measure, affine diameter
Mathematical Subject Classification 2010
Primary: 52A05, 52A20
Secondary: 52A41, 52B55
Milestones
Received: 14 March 2016
Revised: 9 June 2017
Accepted: 9 June 2017
Published: 22 September 2017
Authors
 Qi Guo Department of Mathematics Suzhou University of Science and Technology Suzhou China Gabor Toth Department of Mathematics Rutgers University Camden, NJ United States