Polynomial local functionals on convex functions

Abstract

We show that every continuous local functional on the space of finite convex functions on ${\mathbb{R}}^n$ is a valuation. This relation is used to establish a homogeneous decomposition for the class of polynomial local functionals as well as a classification of translation or rigid motion invariant polynomial local functionals. In addition we discuss implications for the compact-open topology on the space of polynomial local functionals.

37 pages, v2: typos, minor generalization in Section 3 required for the application in arXiv:2605.30927