Zonal valuations on convex bodies

Abstract

A complete classification of all zonal, continuous, and translation invariant valuations on convex bodies is established. The valuations obtained are expressed as principal value integrals with respect to the area measures. The convergence of these principal value integrals is obtained from a new weighted version of an inequality for the volume of spherical caps due to Firey. For Minkowski valuations, this implies a refinement of the convolution representation by Schuster and Wannerer in terms of singular integrals. As a further application, a new proof of the classification of $\mathrm{SO}(n)$-invariant, continuous, and dually epi-translation invariant valuations on the space of finite convex functions by Colesanti, Ludwig, and Mussnig is obtained.