Dual mean Minkowski measures and the Grünbaum conjecture for affine diameters

Qi Guo and Gabor Toth

Vol. 292 (2018), No. 1, 117–137

DOI: 10.2140/pjm.2018.292.117

Abstract

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 $z\in intK$, 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