Variants of a theorem of Macbeath in finite dimensional normed spaces

TL;DR

This paper explores extremal volume properties of convex polytopes inscribed in unit balls across various normed spaces, extending Macbeath's classical Euclidean volume approximation theorem.

Contribution

It introduces normed space variants of Macbeath's theorem, analyzing extremal values of different volume measures for inscribed convex polytopes.

Findings

Identifies extremal volume bounds for polytopes in normed spaces.

Extends classical Euclidean results to general normed spaces.

Provides new insights into volume approximation in convex geometry.

Abstract

A classical theorem of Macbeath states that for any integers $d \geq 2$, $n \geq d+1$, $d$-dimensional Euclidean balls are hardest to approximate, in terms of volume difference, by inscribed convex polytopes with $n$ vertices. In this paper we investigate normed variants of this problem: we intend to find the extremal values of the Busemann volume, Holmes-Thompson volume, Gromov's mass and Gromov's mass$^*$ of a largest volume convex polytope with $n$ vertices, inscribed in the unit ball of a $d$-dimensional normed space.