Crofton Formulae for Tensor-Valued Curvature Measures

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.2010 Mathematics Subject Classification. Primary: 52A20, 53C65; secondary: 52A22, 52A38, 28A75.

Crofton Formulae for Tensorial Curvature Measures: The General Case

“…The aim of the present work is to prove a complete set of Crofton formulae for the (generalized) tensorial curvature measures. This complements the particular results for (extrinsic) tensorial curvature measures and Minkowski tensors obtained in [14] and [29]. The current approach is basically an application of the kinematic formulae for (generalized) tensorial curvature measures derived in [15].…”

“…The second special case concerns the tensorial curvature measures φ r,s,0 k−1 . Although the result has been derived in a different way in our previous work [14,Theorem 4.12], we state it and derive it as a special case of the present more general approach. .…”

“…Comparing Corollary 7 and Corollary 8 to the corresponding results in [14], it should be observed that the normalization of the tensorial measures in [14] is different from the current normalization (although the measures are denoted in the same way).…”

“…This led us to consider their marginal measures on Borel subsets of R n , the tensorial curvature measures and their generalizations on convex polytopes. Preceding this work, the present authors derived a set of Crofton formulae for a different version of these tensorial curvature measures, defined with respect to the (random) intersecting affine subspace, and as a consequence of these results also obtained Crofton formulae for some of the (original) tensorial curvature measures (see [14]). As a far reaching generalization of previous results, a complete set of kinematic formulae for the (generalized) tensorial curvature measures has been proved in [15].…”

“…For easier reference, we state the required kinematic formula, which has recently been proved in [15,Theorem 1]. To state the result, we write G n for the rigid motion group of R n and denote by µ the Haar measure on G n with the usual normalization (see [14], [27, p. 586]).…”