球面上的哈德维格尔定理
The Spherical Hadwiger Theorem
浏览论文内容
中文总结 AI 辅助
作者证明任意维数下的球面哈德维格尔分类定理，给出连续\textit{SO(n+1)}不变赋值的唯一线性组合表示，并通过锥-球面对应推导闭凸锥上的对应分类。
中文摘要 AI 辅助
我们证明了任意维数下的球面上的哈德维格尔分类定理。对于\textit{n≥1}，球面\textit{S^n}中所有闭球面凸集空间上的每个连续\textit{SO(n+1)}不变赋值，都可唯一表示为球面内蕴体积\textit{V_0,…,V_n}的线性组合。该证明采用归纳法，利用了到非真集的唯一延拓、有向球面单形上的连续上循环以及带符号锥变换。通过锥-球面对应，对于\textit{d≥2}，这给出了\textit{R^d}中所有闭凸锥上连续、不一定归一化的\textit{SO(d)}不变锥赋值的对应分类。特别地，两种情形下\textit{SO}不变性均蕴含\textit{O}不变性。
英文摘要
We prove the spherical Hadwiger classification in every dimension. For $n\geq1$, every continuous $SO(n+1)$-invariant valuation on the space of all closed spherical convex sets in $\mathbb{S}^n$ can be written uniquely as a linear combination of the spherical intrinsic volumes $V_0,\ldots,V_n$. The same classification holds when the domain is restricted to sets contained in an open hemisphere. The proof is inductive: it reduces the problem to a vanishing statement for simple valuations and uses a continuous alternating cocycle on tuples of spherical points together with a signed coning transform. Through the cone--sphere correspondence, this yields, for $d\geq2$, the corresponding classification of continuous $SO(d)$-invariant conic valuations on all closed convex cones in $\mathbb{R}^d$, without imposing normalization at the zero cone. In particular, $SO$-invariance implies $O$-invariance in both settings.
发表机构
- School of Mathematics, Hunan University（湖南大学数学学院）
机构由 AI 辅助整理，请以论文原文为准。
