The Spherical Hadwiger Theorem

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 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.