From valuations on convex bodies to convex functions

Abstract

A geometric framework relating valuations on convex bodies to valuations on convex functions is introduced. It is shown that a classical result by McMullen can be used to obtain a characterization of continuous, epi-translation invariant, and n-epi-homogeneous valuations on convex functions, which was previously established by Colesanti, Ludwig, and Mussnig. Following an approach by Goodey and Weil, a new characterization of 1-epi-homogeneous valuations is obtained.

arXiv commentA new section was added after the Preliminaries, adding information on the methods and connections with the literature. Abstract, Introduction, and Preliminaries have been restructured