Local Tensor Valuations

Hug, Daniel; Schneider, Rolf

doi:10.1007/s00039-014-0289-0

Cited by 34 publications

(79 citation statements)

References 48 publications

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“…Surprising new insight into integral geometric formulae can be gained by combining local and tensorial extensions of the classical intrinsic volumes. This setting has recently been studied by Schneider (see [66]) and further analyzed by Hug & Schneider in their works on local tensor valuations (see [40,41,42]). These valuations can be viewed as tensor-valued generalizations of the support measures.…”

Section: Introductionmentioning

confidence: 99%

Kinematic formulae for tensorial curvature measures

Hug

Weis

2018

Annali di Matematica

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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.

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Section: Introductionmentioning

confidence: 99%

Kinematic formulae for tensorial curvature measures

Hug

Weis

2018

Annali di Matematica

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“…In this section, we show that the generalized tensorial curvature measures multiplied with powers of the metric tensor are linearly independent. The proof of this result follows the argument for Theorem 3.1 in [9]. In particular, Theorem 10 shows that the Crofton formulae, which we stated in Section 3 and proved in Section 4, cannot be simplified further, as there are no more linear dependences between the appearing functionals.…”

Section: Linear Independence Of the Generalized Tensorial Curvature Mmentioning

confidence: 66%

“…We emphasize that in the present work, φ r,s,l j (P, ·) and φ r,0,l n (K, ·) are Borel measures on R n and not on R n × S n−1 , as in [9], and also the normalization is slightly adjusted as compared to [9] (where the normalization was not a relevant issue). However, we stick to the definition and normalization of our preceding work [15], where the connection to the generalized local Minkowski tensorsφ r,s,l j is described and where also the properties and available characterization results for these measures are discussed in more detail.…”

Section: Preliminariesmentioning

confidence: 99%

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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“…Tensor valuations on convex bodies have attracted increasing attention in recent years (see, e.g., [7,23,26]). They were introduced by McMullen in [37] and Alesker subsequently obtained a complete classification of continuous and isometry equivariant tensor valuations on convex bodies (based on [3] but completed in [4]).…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Tensor valuations on lattice polytopes

Ludwig

Silverstein

2017

Advances in Mathematics

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The Ehrhart polynomial and the reciprocity theorems by Ehrhart & Macdonald are extended to tensor valuations on lattice polytopes. A complete classification is established of tensor valuations of rank up to eight that are equivariant with respect to the special linear group over the integers and translation covariant. Every such valuation is a linear combination of the Ehrhart tensors which is shown to no longer hold true for rank nine. 2010

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Local Tensor Valuations

Cited by 34 publications

References 48 publications

Kinematic formulae for tensorial curvature measures

Kinematic formulae for tensorial curvature measures

Crofton Formulae for Tensorial Curvature Measures: The General Case

Tensor valuations on lattice polytopes

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