arXiv subjects

Jan A. Weis

Publications and source records attributed to Jan A. Weis.

4 recordsLinked to original sources

arXiv2017

Integral geometric formulae for Minkowski tensors

The Minkowski tensors are the natural tensor-valued generalizations of the intrinsic volumes of convex bodies. We prove two complete sets of integral geometric formulae, so called kinematic and Crofton formulae, for these Minkowski tensors. These formulae express the integral mean of the Minkowski tensors of the intersection of a given convex body with a second geometric object (another convex body in the kinematic case and an affine subspace in the Crofton case) which is uniformly moved by a proper rigid motion, in terms of linear combinations of the Minkowski tensors of the given geometric objects.

math.MG↗

arXiv2017

Crofton formulae for tensorial curvature measures: the general case

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.

math.MG↗

arXiv2016

Kinematic formulae for tensorial curvature measures

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.

math.MG↗

arXiv2016

Crofton Formulae for Tensor-Valued Curvature Measures

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.

math.MG↗