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A Klain-Schneider Theorem for Vector-Valued Valuations on Convex Functions

Mohamed A. Mouamine, Fabian Mussnig

TL;DR

The paper extends the Klain–Schneider framework to vector-valued valuations on convex functions, providing a complete classification of continuous, translation covariant, simple valuations and revealing a dual set of functional operators that correspond to classical geometric invariants. It introduces two families of functionals—functional intrinsic moments and Minkowski vectors—via Steiner-type formulas, Monge–Ampère measures, and Hessian measures, and shows how rotation (or O(n) in low dimensions) equivariance yields canonical decompositions into top-degree and moment-type terms. The results unify and extend the functional Hadwiger program, linking valuations on convex functions to intrinsic volumes, Monge–Ampère theory, and Hessian-measure calculus, while producing genuinely new Minkowski-vector operators absent in the classical body setting. Together with homogeneous-valuation classifications, the work offers a robust framework for understanding how convex-function valuations behave under translations, rotations, and dilations, with potential implications for analysis on convex-frontier function spaces and related geometric measure theory.

Abstract

A functional analog of the Klain-Schneider theorem for vector-valued valuations on convex functions is established, providing a classification of continuous, translation covariant, simple valuations. Under additional rotation equivariance assumptions, an analytic counterpart of the moment vector is characterized alongside a new epi-translation invariant valuation. The former arises as the top-degree operator in a family of functional intrinsic moments, which are linked to functional intrinsic volumes through translations. The latter represents the top-degree operator in a class of Minkowski vectors, which are introduced in this article and which lack classical counterparts on convex bodies, as they vanish due to the Minkowski relations. Additional classification results are obtained for homogeneous valuations of extremal degrees.

A Klain-Schneider Theorem for Vector-Valued Valuations on Convex Functions

TL;DR

The paper extends the Klain–Schneider framework to vector-valued valuations on convex functions, providing a complete classification of continuous, translation covariant, simple valuations and revealing a dual set of functional operators that correspond to classical geometric invariants. It introduces two families of functionals—functional intrinsic moments and Minkowski vectors—via Steiner-type formulas, Monge–Ampère measures, and Hessian measures, and shows how rotation (or O(n) in low dimensions) equivariance yields canonical decompositions into top-degree and moment-type terms. The results unify and extend the functional Hadwiger program, linking valuations on convex functions to intrinsic volumes, Monge–Ampère theory, and Hessian-measure calculus, while producing genuinely new Minkowski-vector operators absent in the classical body setting. Together with homogeneous-valuation classifications, the work offers a robust framework for understanding how convex-function valuations behave under translations, rotations, and dilations, with potential implications for analysis on convex-frontier function spaces and related geometric measure theory.

Abstract

A functional analog of the Klain-Schneider theorem for vector-valued valuations on convex functions is established, providing a classification of continuous, translation covariant, simple valuations. Under additional rotation equivariance assumptions, an analytic counterpart of the moment vector is characterized alongside a new epi-translation invariant valuation. The former arises as the top-degree operator in a family of functional intrinsic moments, which are linked to functional intrinsic volumes through translations. The latter represents the top-degree operator in a class of Minkowski vectors, which are introduced in this article and which lack classical counterparts on convex bodies, as they vanish due to the Minkowski relations. Additional classification results are obtained for homogeneous valuations of extremal degrees.

Paper Structure

This paper contains 11 sections, 23 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. The Klain--Schneider Theorem
  3. Real-Valued Valuations on Convex Functions
  4. Vector-Valued Valuations on Convex Functions
  5. Plan of This Article
  6. Preliminaries
  7. Classification of Simple Valuations
  8. Functional Intrinsic Moments and Minkowski Vectors
  9. Classification of Homogeneous Valuations
  10. Homogeneity of Degree $\boldsymbol{0}$
  11. Homogeneity of Top Degrees

Key Result

Theorem 1.1

A map $\operatorname{Z}\colon\mathcal{K}^n\to\mathbb{R}$ is a continuous, translation invariant, simple valuation, if and only if there exist a constant $c\in\mathbb{R}$ and an odd, continuous function $g\colon\mathbb{S}^{n-1}\to\mathbb{R}$ such that for every $K\in\mathcal{K}^n$.

Theorems & Definitions (34)

  • Theorem 1.1: Klain--Schneider
  • Corollary 1.2
  • Theorem 1.3: Klain--Schneider theorem on ${\mathrm{Conv}(\mathbb{R}^n; \mathbb{R})}$
  • Theorem 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Proposition 3.1
  • ...and 24 more