Valuations on convex functions and convex sets and Monge-Ampere operators

Abstract

The notion of a valuation on convex bodies is very classical. The notion of a valuation on a class of functions was recently introduced and studied by M. Ludwig and others. We study an explicit relation between continuous valuations on convex functions which are invariant under adding arbitrary linear functionals, and translations invariant continuous valuations on convex bodies. More precisely, we construct a natural linear map from the former space to the latter and prove that it has dense image and infinite dimensional kernel. The proof uses the author's irreducibility theorem and few properties of the real Monge-Ampere operators due to A.D. Alexandrov and Z. Blocki. Fur- thermore we show how to use complex, quaternionic, and octonionic Monge-Ampere operators to construct more examples of continuous valuations on convex functions in an analogous way.