  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.
