A Characterization of Functional Affine Surface Areas

Abstract

A characterization of valuations on the space of convex Lipschitz functions whose domain is a polytope in ${\mathbb{R}}^n$ is obtained. It is shown that every upper semicontinuous, equi-affine and dually epi-translation invariant valuation can be written as a linear combination of a constant term, the volume of the domain, and a functional affine surface area. In addition, dual statements for finite-valued convex functions are established.