# On the magnitude and intrinsic volumes of a convex body in Euclidean space

## TL;DR

This paper establishes an upper bound for the magnitude of convex bodies in Euclidean space based on intrinsic volumes, linking geometric invariants with metric space properties and extending to infinite-dimensional contexts.

## Contribution

It introduces a new upper bound for the magnitude of convex bodies in Euclidean space using intrinsic volumes, derived via embeddings into high-dimensional  spaces.

## Key findings

- Upper bound for magnitude in terms of intrinsic volumes
- Condition for finite magnitude of infinite-dimensional subsets
- Sharpness of the bound for Euclidean balls shrinking to a point

## Abstract

Magnitude is an isometric invariant of metric spaces inspired by category theory. Recent work has shown that the asymptotic behavior under rescaling of the magnitude of subsets of Euclidean space is closely related to intrinsic volumes. Here we prove an upper bound for the magnitude of a convex body in Euclidean space in terms of its intrinsic volumes. The result is deduced from an analogous known result for magnitude in $\ell_1^N$, via approximate embeddings of Euclidean space into high-dimensional $\ell_1^N$ spaces. As a consequence, we deduce a sufficient condition for infinite-dimensional subsets of a Hilbert space to have finite magnitude. The upper bound is also shown to be sharp to first order for an odd-dimensional Euclidean ball shrinking to a point; this complements recent work investigating the asymptotics of magnitude for large dilatations of sets in Euclidean space.