Minkowski valuations on convex functions

TL;DR

This paper classifies and characterizes Minkowski valuations on convex functions, including the projection body operator and a new covariant valuation, enriching the understanding of geometric measures in convex analysis.

Contribution

It provides a comprehensive classification of SL(n) contravariant Minkowski valuations and introduces a novel SL(n) covariant valuation on convex functions.

Findings

Classification of SL(n) contravariant Minkowski valuations

Characterization of the projection body operator

Introduction of a new SL(n) covariant Minkowski valuation

Abstract

A classification of SL$(n)$ contravariant Minkowski valuations on convex functions and a characterization of the projection body operator are established. The associated LYZ measure is characterized. In addition, a new SL$(n)$ covariant Minkowski valuation on convex functions is defined and characterized.