Variants of a theorem of Macbeath in finite dimensional normed spaces

2025/07/15 by Lángi, Z., Wang, S.
#52A21 #52A27 #52A40 #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2507.11496

Abstract

A classical theorem of Macbeath states that for any integers d ≥ 2, n ≥ 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.

Citations

• On the Vertices of Delta-modular Polyhedra
• Steiner symmetrization on the sphere
• Polytopes of Maximal Volume Product
• On a normed version of a Rogers-Shephard type problem
• Shadow systems and volume of polar convex bodies
• Equiframed Curves ? A Generalization of Radon Curves

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