Kinematic formulae for tensorial curvature measures

Abstract

Tensorial curvature measures are tensor-valued generalizations of the curvature measures of convex bodies. We prove a complete set of kinematic formulae for such tensorial curvature measures on convex bodies and for their (nonsmooth) generalizations on convex polytopes. These formulae express the integral mean of the tensorial curvature measure of the intersection of two given convex bodies (resp. polytopes), one of which is uniformly moved by a proper rigid motion, in terms of linear combinations of tensorial curvature measures of the given convex bodies (resp. polytopes). We prove these results in a more direct way than in the classical proof of the principal kinematic formula for curvature measures, which uses the connection to Crofton formulae to determine the involved constants explicitly.Date: December 26, 2016. 2010 Mathematics Subject Classification. Primary: 52A20, 53C65; secondary: 52A22, 52A38, 28A75.

Crofton Formulae for Tensorial Curvature Measures: The General Case

“…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 connection between local kinematic and local Crofton formulae is well known for the scalar curvature measures.…”

“…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 the determination of these kinds of geometric features, rotation covariant valuations, such as the Minkowski tensors and the (generalized) tensorial curvature measures considered here, are much more suitable, and therefore they have been heavily used recently (see for example [28]). For further applications we refer to the introduction of our preceding work [15].…”

“…In the tensorial framework, this approach breaks down, since the explicit calculation of integral mean values of (generalized) tensorial curvature measures for sufficiently many examples (template method) does not seem to be possible. Instead, the required coefficients were determined by a direct derivation of the kinematic formulae for (generalized) tensorial curvature measures in [15], and from this we now can derive explicit Crofton formulae, otherwise following the reasoning described above.…”

Intersection probabilities and kinematic formulas for polyhedral cones

“…For instance, it seems to be inefficient for the extension of kinematic formulas to the tensor-valued generalizations of the intrinsic volumes. For these, the existing proofs in the Euclidean case are still rather complicated, see [10] and [11]. For that reason, approaches to local kinematic formulas by direct computation should be given more attention.…”