Variants of a theorem of Macbeath in finite‐dimensional normed spaces

A classical theorem of Macbeath states that for any integers , , ‐dimensional Euclidean balls are hardest to approximate, in terms of volume difference, by inscribed convex polytopes with 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 of a largest volume convex polytope with vertices, inscribed in the unit ball of a ‐dimensional normed space.