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Additive kinematic formulas for convex functions

Daniel Hug, Fabian Mussnig, Jacopo Ulivelli

TL;DR

This work develops a functional analogue of the additive kinematic formula by marrying Hadwiger-type results for convex functions with a Kubota-type formula for conjugate Monge–Ampère measures. The authors introduce functional intrinsic volumes $\overline{V}_{j,\alpha}^*$ on convex functions and prove a precise, $SO(n)$-averaged identity that expresses these functionals in terms of conjugate MA measures, revealing a unifying framework that connects Hessian-integral representations to MA-measure representations. They further obtain functional analogues of mixed volumes, discuss inf-deconvolution, and derive a new integral formula for mixed area measures over $SO(n-1)\times O(1)$, with applications to convex bodies via floor maps and gnomonic projections. The results illuminate the deep link between convex-analytic valuations, Monge–Ampère theory, and integral geometry, providing a robust toolkit for functional and geometric generalizations of classical intrinsic volumes. Overall, the paper advances our understanding of how functional-analytic and geometric invariants interact through MA theory and rotational symmetry.

Abstract

We prove a functional version of the additive kinematic formula as an application of the Hadwiger theorem on convex functions together with a Kubota-type formula for mixed Monge-Ampère measures. As an application, we give a new explanation for the equivalence of the representations of functional intrinsic volumes as singular Hessian valuations and as integrals with respect to mixed Monge-Ampère measures. In addition, we obtain a new integral geometric formula for mixed area measures of convex bodies, where integration on $\operatorname{SO}(n-1)\times \operatorname{O}(1)$ is considered.

Additive kinematic formulas for convex functions

TL;DR

This work develops a functional analogue of the additive kinematic formula by marrying Hadwiger-type results for convex functions with a Kubota-type formula for conjugate Monge–Ampère measures. The authors introduce functional intrinsic volumes on convex functions and prove a precise, -averaged identity that expresses these functionals in terms of conjugate MA measures, revealing a unifying framework that connects Hessian-integral representations to MA-measure representations. They further obtain functional analogues of mixed volumes, discuss inf-deconvolution, and derive a new integral formula for mixed area measures over , with applications to convex bodies via floor maps and gnomonic projections. The results illuminate the deep link between convex-analytic valuations, Monge–Ampère theory, and integral geometry, providing a robust toolkit for functional and geometric generalizations of classical intrinsic volumes. Overall, the paper advances our understanding of how functional-analytic and geometric invariants interact through MA theory and rotational symmetry.

Abstract

We prove a functional version of the additive kinematic formula as an application of the Hadwiger theorem on convex functions together with a Kubota-type formula for mixed Monge-Ampère measures. As an application, we give a new explanation for the equivalence of the representations of functional intrinsic volumes as singular Hessian valuations and as integrals with respect to mixed Monge-Ampère measures. In addition, we obtain a new integral geometric formula for mixed area measures of convex bodies, where integration on is considered.
Paper Structure (6 sections, 21 theorems, 111 equations)

This paper contains 6 sections, 21 theorems, 111 equations.

Table of Contents

  1. Introduction and Statement of Results
  2. Preliminaries
  3. Proof of Theorem \ref{['thm:add_kin_form_ma']}
  4. Further Formulas
  5. Singular Hessian Integrals
  6. Formulas for Convex Bodies

Key Result

Theorem 1.1

A map $\mathop{\mathrm{\operatorname{Z}}}\nolimits\colon {\mathcal{K}}^n\to{\mathbb R}$ is a continuous, translation and rotation invariant valuation if and only if there exist $c_0,\ldots,c_n\in{\mathbb R}$ such that for $K\in{\mathcal{K}}^n$.

Theorems & Definitions (28)

  • Theorem 1.1: Hadwiger's Theorem
  • Theorem 1.2: Additive Kinematic Formula
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • ...and 18 more