Monge–Ampère operators and valuations

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 \(\mathbb {R}^n\) ${\mathbb{R}}^n$ 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 \({{\,\mathrm{\textrm{SO}}\,}}(n)\) $\text{SO}(n)$ -equivariant valuations in terms of Hessian measures.