Smooth valuations on convex functions

TL;DR

This paper develops a new class of valuations on convex functions using differential forms and explores their properties, including density and invariance under certain group actions.

Contribution

It introduces a construction of smooth valuations on convex functions via differential forms and characterizes their density and invariance properties.

Findings

Constructed valuations via differential forms on convex functions.

Described the kernel of the valuation construction.

Established density of smooth valuations in the space of continuous valuations.

Abstract

We construct valuations on the space of finite-valued convex functions using integration of differential forms over the differential cycle associated to a convex function. We describe the kernel of this procedure and show that the intersection of this space of smooth valuations with the space of all continuous dually epi-translation invariant valuations on convex functions is dense in the latter. As an application, we obtain a description of 1-homogeneous, continuous, dually epi-translation invariant valuations that are invariant with respect to a compact subgroup operating transitively on the unit sphere.