The Spherical Hadwiger Theorem

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 ${\text{Sphere}}^n$ can be written uniquely as a linear combination of the spherical intrinsic volumes $V_0,\dots ,V_n$. 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 $d\ge 2$, the corresponding classification of continuous, not necessarily normalized, $SO(d)$-invariant conic valuations on all closed convex cones in ${\mathbb{R}}^d$. In particular, $SO$-invariance implies $O$-invariance in both settings.