Valuations on convex functions and convex sets and Monge–Ampère operators

Abstract

Abstract The notion of a valuation on convex bodies is very classical; valuations on a class of functions have been 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 translation 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 properties of the real Monge–Ampère operators due to A.D. Alexandrov and Z. Blocki. Furthermore we show how to use complex, quaternionic, and octonionic Monge–Ampère operators to construct more examples of continuous valuations on convex functions in an analogous way.