AI 中文总结
该研究证明球多面体上连续且等距不变的赋值是球内蕴体积的线性组合，借助多面体赋值的弱可微性，推广Alesker的拟光滑赋值结果并简化问题至平移不变情形。
AI 中文摘要
我们证明了球多面体上每个连续且等距不变的赋值，都是球内蕴体积的线性组合。该证明依赖于$\boldsymbol{\text{R}}^n$中多面体上赋值所满足的弱可微性性质，该性质具有关于仿射群作用的自然光滑性。这使我们能将Alesker针对拟光滑赋值建立的若干结果推广到多面体情形，并将该问题简化为多面体上可测赋值的平移不变情形。
英文摘要
We show that every continuous and isometry invariant valuation on spherical polytopes is a linear combination of the spherical intrinsic volumes. The proof relies on a weak differentiability property satisfied by valuations on polytopes in $\mathbb{R}^n$ with a natural smoothness property with respect to the action of the affine group. This enables us to transfer several results established by Alesker for quasi-smooth valuations to the polytopal setting, and to reduce the problem to the translation invariant case of measurable valuations on polytopes.
Comments20 pages
