Polynomial local functionals on convex functions

TL;DR

This paper proves that continuous local functionals on convex functions are valuations, and classifies polynomial local functionals, revealing their structure and invariance properties in convex analysis.

Contribution

It establishes that all continuous local functionals are valuations and provides a classification of polynomial local functionals based on invariance and decomposition.

Findings

Every continuous local functional is a valuation.

Polynomial local functionals admit a homogeneous decomposition.

Classification of invariant polynomial local functionals was achieved.

Abstract

We shown 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.