Hadwiger's classification theorem on the sphere via signed orthoscheme decompositions

We give an elementary and combinatorial proof of the spherical Hadwiger theorem, which states that every continuous, rotation-invariant valuation on the space of closed convex cones in ${\mathbb{R}}^n$ is a linear combination of the conic intrinsic volumes. Our proof uses a signed orthoscheme decomposition. Its vanishing step requires only integrability of an orthoscheme restriction of the residual valuation, a weaker regularity condition than continuity.