A geometric decomposition for unitarily invariant valuations on convex functions

TL;DR

This paper classifies and decomposes unitarily invariant valuations on convex functions in complex space, revealing a structure akin to hermitian intrinsic volumes and providing integral representations involving Monge-Ampère operators.

Contribution

It provides a complete classification and geometric decomposition of these valuations, introducing new integral representations and linking to hermitian intrinsic volumes.

Findings

Decomposition of valuations into subspaces based on vanishing properties

Representation of valuations via principal value integrals

Connection to hermitian intrinsic volumes

Abstract

Valuations on the space of finite-valued convex functions on $\mathbb{C}^n$ that are continuous, dually epi-translation invariant, as well as $\mathrm{U}(n)$-invariant are completely classified. It is shown that the space of these valuations decomposes into a direct sum of subspaces defined in terms of vanishing properties with respect to restrictions to a finite family of special subspaces of $\mathbb{C}^n$, mirroring the behavior of the hermitian intrinsic volumes introduced by Bernig and Fu. Unique representations of these valuations in terms of principal value integrals involving two families of Monge-Amp\`ere-type operators are established