A geometric decomposition for unitarily invariant valuations on convex functions

Abstract

Valuations on the space of finite-valued convex functions on ℂⁿ that are continuous, dually epi-translation invariant, as well as 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 ℂⁿ, 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-Ampère-type operators are established.