The Spherical Hadwiger Theorem

Abstract

We prove the spherical Hadwiger classification in every dimension. For $n\ge 1$, 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 $V_0,\dots ,V_n$. 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 $d\ge 2$, 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.

26 pages