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Averkov, Gennadiy : On the inequality for volume and Minkowskian thickness

Averkov, Gennadiy : On the inequality for volume and Minkowskian thickness

Author(s):
Averkov, Gennadiy
Title:
On the inequality for volume and Minkowskian thickness
Preprint series:
Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 2, 2004
Mathematics Subject Classification:
52A21 [ Finite-dimensional Banach spaces ]
52A10 [ Convex sets in $2$ dimensions ]
52A38 [ Length, area, volume ]
Abstract:
Let $K$ be a convex body in a $d$-dimensional Minkowski space ($=$ real Banach space of dimension $d$) with unit ball $B.$ The Minkowskian thickness $\triangle_B(K)$ of $K$ is the minimal Minkowskian distance occurring between two points lying in different parallel supporting hyperplanes of $K.$ The relation between volume $V(K)$ of $K$ and the Minkowskian thickness of $K$ is obviously given by the sharp geometric inequality $V(K) \ge \alpha(B) \cdot \triangle_B(K)^d $ with some positive coefficient $\alpha(B)$ depending on the space. We prove that $\binom{2d}{d}^{-1} \le \alpha(B)/V(B) \le 2^{-d}$ with $\alpha(B)/V(B)=\binom{2d}{d}^{-1}$ if and only if $B$ is the difference body of a simplex and $\alpha(B)/V(B)=2^{-d}$ if (and, provided $d=2,$ also only if) $B$ is cross-polytope. The question whether for $d \ge 3$ the condition $\alpha(B)/V(B)=2^{-d}$ implies that $B$ is a cross-polytope remains open.
Keywords:
Blaschke-Lebesgue theorem, convexity, cross-section measure, difference body, finite dimensional Banach space, minimum width, Minkowski space, thickness, width, width function, reduced body, geometric inequalities
Language:
English
Publication time:
3 / 2004