On the Inequality for Volume and Minkowskian Thickness

Averkov, Gennadiy

Canadian mathematical bulletin, Tome 49 (2006) no. 2, pp. 185-195

Résumé

Given a centrally symmetric convex body $B$ in ${{\mathbb{E}}^{d}}$ , we denote by ${{\mathcal{M}}^{d}}\left( B \right)$ the Minkowski space (i.e., finite dimensional Banach space) with unit ball $B$ . Let $K$ be an arbitrary convex body in ${{\mathcal{M}}^{d}}\left( B \right)$ . The relationship between volume $V\left( K \right)$ and the Minkowskian thickness (= minimal width) ${{\Delta }_{B}}\left( K \right)$ of $K$ can naturally be given by the sharp geometric inequality $V\left( K \right)\ge \alpha \left( B \right)\cdot {{\Delta }_{B}}{{\left( K \right)}^{d}}$ , where $\alpha \left( B \right)>0$ . As a simple corollary of the Rogers-Shephard inequality we obtain that ${{\left( _{d}^{2d} \right)}^{-1}}\le \alpha \left( B \right)/V\left( B \right)\le {{2}^{-d}}$ with equality on the left attained if and only if $B$ is the difference body of a simplex and on the right if $B$ is a cross-polytope. The main result of this paper is that for $d=2$ the equality on the right implies that $B$ is a parallelogram. The obtained results yield the sharp upper bound for the modified Banach–Mazur distance to the regular hexagon.

Détail

DOI : 10.4153/CMB-2006-019-4

Mots-clés : 52A40, 46B20, Convex body, geometric inequality, thickness, Minkowski space, Banach space, normed space, reduced body, Banach-Mazur compactum, (modified) Banach-Mazur distance, volume ratio