Singular valuations and the Hadwiger theorem on convex functions

TL;DR

This paper characterizes certain valuations on convex functions, offers a new proof of the Hadwiger theorem in this context, and introduces novel integral representations of functional intrinsic volumes.

Contribution

It provides a new characterization of valuations invariant under specific transformations and a novel proof of the Hadwiger theorem for convex functions.

Findings

Characterization of smooth, rotation, and dually epi-translation invariant valuations

New proof of the Hadwiger theorem on convex functions

Representation of functional intrinsic volumes as principal value integrals

Abstract

We give a characterization of smooth, rotation and dually epi-translation invariant valuations and use this result to obtain a new proof of the Hadwiger theorem on convex functions. We also give a description of the construction of the functional intrinsic volumes using integration over the differential cycle and provide a new representation of these functionals as principal value integrals with respect to the Hessian measures.