Polynomial valuations on convex functions and their maximal extensions

TL;DR

This paper investigates extension problems for polynomial valuations on convex functions, revealing geometric obstructions and providing explicit integral representations for top-degree valuations, based on a homogeneous decomposition approach.

Contribution

It introduces a new geometric criterion for extending polynomial valuations and derives explicit integral formulas for valuations of top degree.

Findings

Extension problems reduce to geometric obstructions.

Homogeneous decomposition is key to analysis.

Explicit integral representations are established.

Abstract

Extension problems for polynomial valuations on different cones of convex functions are investigated. It is shown that for the classes of functions under consideration, the extension problem reduces to a simple geometric obstruction on the support of these valuations. The results rely on a homogeneous decomposition for the space of polynomial valuations of bounded degree and the support properties of certain distributions associated to the homogeneous components. As an application, an explicit integral representation for valuations of top degree is established.