A Characterization of Functional Affine Surface Areas

TL;DR

This paper characterizes valuations on convex Lipschitz functions over polytopes, showing they can be expressed as combinations of volume, a constant, and a functional affine surface area, with dual results for convex functions.

Contribution

It provides a complete characterization of certain valuations on convex functions, linking them to affine surface areas and establishing dual statements.

Findings

Valuations can be decomposed into volume, constant, and affine surface area components.

Upper semicontinuous, equi-affine, and dually epi-translation invariant valuations are characterized.

Dual results for finite-valued convex functions are established.

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.