Magnitude and Holmes-Thompson intrinsic volumes of convex bodies

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.

v3: minor expositional clarifications. To appear in the Canadian Mathematical Bulletin

References in corpus (2)

• The Willmore energy and the magnitude of Euclidean domains
• Further inequalities for the (generalized) Wills functional