Unitarily invariant valuations on convex functions

TL;DR

This paper characterizes continuous, unitarily invariant valuations on convex functions in complex space, providing integral representations, uniqueness results, and how these valuations are determined by their behavior on subspaces.

Contribution

It introduces a novel integral representation for smooth valuations, establishes uniqueness of valuations via subspace restrictions, and connects these to Fourier-Laplace transforms of Goodey-Weil distributions.

Findings

Integral representation of smooth valuations using Monge-Ampère-type operators

Homogeneous valuations are determined by restrictions to subspaces

Valuations are uniquely identified by finite subspace restrictions

Abstract

Continuous, dually epi-translation invariant valuations on the space of finite-valued convex functions on $\mathbb{C}^n$ that are invariant under the unitary group are investigated. It is shown that elements belonging to the dense subspace of smooth valuations admit a unique integral representation in terms of two families of Monge-Amp\`ere-type operators. In addition, it is proved that homogeneous valuations are uniquely determined by restrictions to subspaces of appropriate dimension and that this information is encoded in the Fourier-Laplace transform of the associated Goodey-Weil distributions. These results are then used to show that a continuous unitarily invariant valuation is uniquely determined by its restriction to a certain finite family of subspaces of $\mathbb{C}^n$.