Variants of a theorem of Macbeath in finite dimensional normed spaces

Z. Lángi; S. Wang

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arXiv · 2507.11496

Variants of a theorem of Macbeath in finite dimensional normed spaces

Z. Lángi·S. Wang

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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.

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BibTeX RIS

Z. Lángi, S. Wang. 2025-07-15. Variants of a theorem of Macbeath in finite dimensional normed spaces. https://arxiv.org/abs/2507.11496

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