The Vectorial Hadwiger Theorem on Convex Functions

Abstract

A complete classification of continuous, dually epi-translation invariant, and rotation equivariant valuations on convex functions is established. This characterizes the recently introduced functional Minkowski vectors, which naturally extend the classical Minkowski relations. For this, the existence of these operators with singular densities is shown, along with additional representations involving mixed Monge-Amp\`ere measures, Kubota-type formulas, and area measures of higher dimensional convex bodies. Dual results are formulated for valuations on super-coercive convex functions.