Unitarily invariant valuations on convex functions

Abstract

Continuous, dually epi-translation invariant valuations on the space of finite-valued convex functions on $C^n$ 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–Ampère-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 $C^n$.