Crofton Formulae for Tensor-Valued Curvature Measures

Hug, Daniel; Weis, Jan A.

doi:10.48550/arxiv.1606.05131

Cited by 3 publications

(18 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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].…”

Section: Introductionsupporting

confidence: 69%

“…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. and k, r, s ∈ N…”

Section: The Crofton Formulaementioning

confidence: 87%

“…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).…”

Section: The Crofton Formulaementioning

confidence: 91%

“…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].…”

Section: Introductionmentioning

confidence: 93%

“…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]).…”

Section: 1mentioning

confidence: 99%

See 4 more Smart Citations

Crofton Formulae for Tensorial Curvature Measures: The General Case

Hug

Weis

2017

Analytic Aspects of Convexity

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Introductionsupporting

confidence: 69%

Section: The Crofton Formulaementioning

confidence: 87%

Section: The Crofton Formulaementioning

confidence: 91%

Section: Introductionmentioning

confidence: 93%

Section: 1mentioning

confidence: 99%

See 3 more Smart Citations

Crofton Formulae for Tensorial Curvature Measures: The General Case

Hug

Weis

2017

Analytic Aspects of Convexity

Self Cite

View full text Add to dashboard Cite

show abstract

Rotational Crofton formulae for Minkowski tensors and some affine counterparts

Svane

Jensen

2017

Advances in Applied Mathematics

View full text Add to dashboard Cite

Motivated by applications in local stereology, a new rotational Crofton formula is derived for Minkowski tensors. For sets of positive reach, the formula shows how rotational averages of intrinsically defined Minkowski tensors on sections passing through the origin are related to the geometry of the sectioned set. In particular, for Minkowski tensors of order j − 1 on j-dimensional linear subspaces, we derive an explicit formula for the rotational average involving hypergeometric functions. Sectioning with lines and hyperplanes through the origin is considered in detail. We also study the case where the sections are not restricted to pass through the origin. For sets of positive reach, we here obtain a Crofton formula for the integral mean of intrinsically defined Minkowski tensors on j-dimensional affine subspaces.

show abstract

Integral geometric formulae for Minkowski tensors

Hug,

Weis

2017

Preprint

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Crofton Formulae for Tensor-Valued 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

Rotational Crofton formulae for Minkowski tensors and some affine counterparts

Integral geometric formulae for Minkowski tensors

Contact Info

Product

Resources

About