Translation invariant curvature measures of convex bodies

TL;DR

The paper develops a deep, representation-theoretic framework for translation-invariant curvature measures on convex bodies, introducing a graded, parity-split structure that interacts with GL$(n)$-actions. Central to the approach are the Kiderlen–Weil decomposition and Bernig embedding, which connect curvature measures to translation-invariant valuations and enable a cleaving of their GL$(n)$-module structure. The authors prove the length-2 property for graded components in degrees $0$ and $n-2$, and establish explicit descriptions in degrees $n-1$ and $n$, along with plane-case smoothness results and a robust topological setup. They apply these tools to show curvature measures automatically define probability kernels and to remove the nonnegativity assumption in Schneider–Weil’s characterization of Federer's curvature measures, yielding a sharp, linear-combination description in terms of Federer's basis. Overall, the work clarifies the algebraic and analytic architecture of curvature measures, with implications for stochastic geometry and integral geometry.

Abstract

In a series of papers, Weil initiated the investigation of translation invariant curvature measures of convex bodies, which include as prime examples Federer's curvature measures. In this paper, we continue this line of research by introducing new tools to study curvature measures. Our main results suggest that the space of curvature measures, which is graded by degree and parity, is highly structured: We conjecture that each graded component has length at most $2$ as a representation of the general linear group, and we prove this in degrees $0$ and $n-2$. Beyond this conjectural picture, our methods yield a characterization of Federer's curvature measures under weaker assumptions.

Theorems & Definitions (132)

• Definition 1.1
• Conjecture 1.2
• Conjecture 1.3
• Theorem 1.4
• Corollary 1.5
• Theorem 1.6
• Proposition 1.7
• Theorem 1.8: Schneider--Weil
• Theorem 1.9
• Lemma 2.1