Monotone invariant valuations on convex cones

Abstract

We classify the monotone $\SOn d$-invariant valuations, $d\ge 2$, on the space of all closed convex cones. They are precisely the linear combinations of the conic intrinsic volumes with nondecreasing coefficients, or equivalently, the nonnegative linear combinations of intrinsic volume tails, up to an additive constant. On nonzero pointed cones we obtain the corresponding characterisation by Grassmann angles, settling a conjecture of McMullen. Every such valuation is automatically continuous in the spherical Hausdorff topology and is $\On d$-invariant. Our proof builds on the signed simplex and averaging arguments of Wang and Wu, but does not rely on the continuous classification theorem.

16 pages