Kinematic formulae for tensorial curvature measures

Hug, Daniel; Weis, Jan A.

doi:10.48550/arxiv.1612.08427

Cited by 3 publications

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“…Since the tensorial curvature measures are local versions of the Minkowski tensors, it is rather straightforward to derive Crofton formulae, similar to the ones proved in the present contribution, and apparently also kinematic formulae, similar to the ones obtained in the preceding work [15], for Minkowski tensors as well. This is the subject of the subsequent work [16].…”

Section: Introductionsupporting

confidence: 76%

“…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.…”

Section: Introductionsupporting

confidence: 66%

“…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].…”

Section: Introductionmentioning

confidence: 65%

“…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].…”

Section: Introductionmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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Crofton Formulae for Tensorial Curvature Measures: The General Case

Hug

Weis

2017

Analytic Aspects of Convexity

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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.

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Section: Introductionsupporting

confidence: 76%

Section: Introductionsupporting

confidence: 66%

Section: Introductionmentioning

confidence: 65%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Crofton Formulae for Tensorial Curvature Measures: The General Case

Hug

Weis

2017

Analytic Aspects of Convexity

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Intersection probabilities and kinematic formulas for polyhedral cones

Schneider

2018

Acta Math. Hungar.

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Integral geometric formulae for Minkowski tensors

Hug,

Weis

2017

Preprint

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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.

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Kinematic formulae for tensorial curvature measures

Cited by 3 publications

References 0 publications

Crofton Formulae for Tensorial Curvature Measures: The General Case

Crofton Formulae for Tensorial Curvature Measures: The General Case

Intersection probabilities and kinematic formulas for polyhedral cones

Integral geometric formulae for Minkowski tensors

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