The Vectorial Hadwiger Theorem on Convex Functions
Mohamed A. Mouamine, Fabian Mussnig
TL;DR
The article develops a complete analytic classification of continuous, dually epi-translation invariant, rotation (or O(n)) equivariant valuations on the space ${\rm Conv}(\mathbb{R}^n;\mathbb{R})$ of convex functions, via functional Minkowski vectors $t_{j,\zeta}^*$. It introduces singular-density constructions, connects these functionals to Hessian measures $\Phi_j$ and mixed Monge–Ampère measures $\mathrm{MA}(\cdot)$, and derives Kubota-type formulas and Abel- and Transform-based tools to decompose valuations. A dual theory on super-coercive convex functions is developed, with convex conjugation providing the parallel results. The main result yields a precise, unique representation of any continuous, epi-invariant valuation as a sum of $t_{j,\zeta_j}^*$ with explicitly characterized density classes, supported by a robust framework relating Hessian measures to area measures of higher-dimensional convex bodies. This work extends classical Hadwiger-type theorems to the functional setting, revealing a rich new family of vector-valued invariants intrinsic to convex functions and offering tools for future geometric-analytic applications.
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-Ampère measures, Kubota-type formulas, and area measures of higher dimensional convex bodies. Dual results are formulated for valuations on super-coercive convex functions.