Magnitude and Holmes-Thompson intrinsic volumes of convex bodies

TL;DR

This paper establishes an upper bound for the magnitude of convex bodies in hypermetric normed spaces using Holmes-Thompson intrinsic volumes, leading to new proofs of Mahler's conjecture for zonoids and Sudakov's inequality.

Contribution

It introduces a novel bound linking magnitude and Holmes-Thompson intrinsic volumes for convex bodies in hypermetric spaces, extending previous results.

Findings

Bound for magnitude in terms of Holmes-Thompson volumes

New proof of Mahler's conjecture for zonoids

Simplified proof of Sudakov's minoration inequality

Abstract

Magnitude is a numerical invariant of compact metric spaces, originally inspired by category theory and now known to be related to myriad other geometric quantities. Generalizing earlier results in $\ell_1^n$ and Euclidean space, we prove an upper bound for the magnitude of a convex body in a hypermetric normed space in terms of its Holmes-Thompson intrinsic volumes. As applications of this bound, we give short new proofs of Mahler's conjecture in the case of a zonoid, and Sudakov's minoration inequality.