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This is a Quantitative Study study.

October 10, 2025Open Access

Variants of a theorem of Macbeath in finite dimensional normed spaces

Key Points

The extremal values for the Busemann volume of convex polytopes are determined with respect to normed spaces, signifying a notable result.
Holmes-Thompson volume reveals critical insights into volume differences while analyzing convex polytopes inscribed in unit balls.
Gromov's mass provides new perspectives on the geometric characteristics of polytopes based on their vertex count, influencing mathematical structures.
The research explores variations of the Macbeath theorem, establishing deeper connections between classical and modern geometric theories.

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.

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Cite This Study

Lángi et al. (Tue,) studied this question.

synapsesocial.com/papers/68e861857ef2f04ca37e396b https://doi.org/https://doi.org/10.48550/arxiv.2507.11496