Monge-Ampère operators and valuations

Abstract

Two classes of measure-valued valuations on convex functions related to Monge-Ampère operators are investigated and classified. It is shown that the space of all valuations with values in the space of complex Radon measures on ${\mathbb{R}}^n$ that are locally determined, continuous, dually epi-translation invariant as well as translation equivariant, is finite dimensional. Integral representations of these valuations and a description in terms of mixed Monge-Ampère operators are established, as well as a characterization of $\mathrm{SO}(n)$-equivariant valuations in terms of Hessian measures.

36 pages