Translation invariant curvature measures of convex bodies

TL;DR

This paper advances the mathematical understanding of translation invariant curvature measures of convex bodies, proposing new tools, conjectures, and partial proofs about their structured nature and specific characterizations.

Contribution

It introduces novel methods to analyze the structure of curvature measures and proves key cases, advancing the theoretical framework of convex geometry.

Findings

The space of curvature measures is highly structured and graded.

Confirmed the conjecture in degrees 0 and n-2.

Provided a characterization of Federer's curvature measures under weaker assumptions.

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.