Magnitude and Holmes-Thompson intrinsic volumes of convex bodies

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 l(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.