Variants of a theorem of Macbeath in finite dimensional normed spaces

Abstract

A classical theorem of Macbeath states that for any integers $d\ge 2$, $n\ge d+1$, $d$-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${}^{\ast }$ of a largest volume convex polytope with $n$ vertices, inscribed in the unit ball of a $d$-dimensional normed space.

17 pages, 2 figures