The Spherical Hadwiger Theorem

Suijie Wang; Shengguo Wu

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arXiv · 2608.27305

The Spherical Hadwiger Theorem

Suijie Wang·Shengguo Wu

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Abstract

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 $\Sphere^n$ 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 and uses a unique extension to non-proper sets, a continuous alternating cocycle on oriented spherical simplices, and a signed coning transform. Through the cone--sphere correspondence, this gives, for $d\geq2$, the corresponding classification of continuous, not necessarily normalized, $SO(d)$-invariant conic valuations on all closed convex cones in $\R^d$. In particular, $SO$-invariance implies $O$-invariance in both settings.

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Suijie Wang, Shengguo Wu. 2026-08-27. The Spherical Hadwiger Theorem. https://arxiv.org/abs/2608.27305

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