Zonal valuations on convex bodies

TL;DR

This paper provides a complete classification of zonal, continuous, and translation invariant valuations on convex bodies, expressing them via principal value integrals and refining existing convolution representations.

Contribution

It introduces a new weighted inequality for spherical caps, enabling the convergence analysis of principal value integrals and extending the classification to Minkowski valuations.

Findings

Established convergence of principal value integrals for valuations.

Refined convolution representation for Minkowski valuations.

Provided a new proof for classification of certain invariant valuations on convex functions.

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.