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.

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关于欧氏空间中凸体的大小和内体积

幅度是受范畴论启发的度量空间的等距不变量。最近的工作表明，在欧几里得空间子集的大小重新缩放下的渐近行为与内在体积密切相关。在这里，我们证明了欧几里得空间中凸体在其固有体积方面的大小的上限。结果是通过将欧几里得空间近似嵌入到高维 $\ell_1^N$ 空间中，从 $\ell_1^N$ 中幅度的类似已知结果推导出来的。因此，我们推导出 Hilbert 空间的无限维子集具有有限幅值的充分条件。对于收缩到一个点的奇维欧几里得球，其上限也显示为一阶锐利；