Crofton formulae for tensorial curvature measures: the general case

TL;DR

This paper derives comprehensive Crofton formulae for tensorial curvature measures, extending previous kinematic results and establishing linear independence of these measures on convex polytopes, with simpler coefficients than earlier works.

Contribution

It provides a complete system of Crofton formulae for generalized tensorial curvature measures, simplifying coefficients and proving their linear independence on convex polytopes.

Findings

Derived Crofton formulae express mean tensorial curvature measures of intersections with affine flats.

Coefficients in the formulae are less technical and more transparent than in previous studies.

Proved linear independence of all generalized tensorial curvature measures on convex polytopes.

Abstract

The tensorial curvature measures are tensor-valued generalizations of the curvature measures of convex bodies. On convex polytopes, there exist further generalizations some of which also have continuous extensions to arbitrary convex bodies. In a previous work, we obtained kinematic formulae for all (generalized) tensorial curvature measures. As a consequence of these results, we now derive a complete system of Crofton formulae for such (generalized) tensorial curvature measures. These formulae express the integral mean of the (generalized) tensorial curvature measures of the intersection of a given convex body (resp. polytope, or finite unions thereof) with a uniform affine $k$-flat in terms of linear combinations of (generalized) tensorial curvature measures of the given convex body (resp. polytope, or finite unions thereof). The considered generalized tensorial curvature measures generalize those studied formerly in the context of Crofton-type formulae, and the coefficients involved in these results are substantially less technical and structurally more transparent than in previous works. Finally, we prove that essentially all generalized tensorial curvature measures on convex polytopes are linearly independent. In particular, this implies that the Crofton formulae which we prove in this contribution cannot be simplified further.