  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.
