Crofton Formulae for Tensor-Valued Curvature Measures

TL;DR

This paper establishes Crofton formulae for tensor-valued curvature measures of convex bodies, linking integrals over affine flats to curvature measures, and simplifies constants in Minkowski tensor calculations.

Contribution

It introduces Crofton formulae for tensorial curvature measures and simplifies related constants, extending previous Minkowski tensor results.

Findings

Derived Crofton formulae for tensorial curvature measures.

Simplified constants in Minkowski tensor calculations.

Extended results to ambient Euclidean space context.

Abstract

The tensorial curvature measures are tensor-valued generalizations of the curvature measures of convex bodies. We prove a set of Crofton formulae for such tensorial curvature measures. These formulae express the integral mean of the tensorial curvature measures of the intersection of a given convex body with a uniform affine $k$-flat in terms of linear combinations of tensorial curvature measures of the given convex body. Here we first focus on the case where the tensorial curvature measures of the intersection of the given body with an affine flat is defined with respect to the affine flat as its ambient space. From these formulae we then deduce some new and also recover known special cases. In particular, we substantially simplify some of the constants that were obained in previous work on Minkowski tensors. In a second step, we explain how the results can be extended to the case where the tensorial curvature measure of the intersection of the given body with an affine flat is determined with respect to the ambient Euclidean space.