Integral Geometry and Algebraic Structures for Tensor Valuations

Bernig, Andreas; Hug, Daniel

doi:10.1007/978-3-319-51951-7_4

Cited by 5 publications

(5 citation statements)

References 12 publications

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“…The algebra of symmetric tensors over R n is denoted by T (the underlying R n will be clear from the context), the vector space of symmetric tensors of rank p ∈ N 0 is denoted by T p with T 0 = R. The symmetric tensor product of tensors T i ∈ T, i = 1, 2, over R n is denoted by T 1 T 2 ∈ T, and for q ∈ N 0 and a tensor T ∈ T we write T q for the q-fold tensor product of T , where T 0 := 1; see also the contributions [43,16] in the lecture notes [49] for further details and references. Identifying R n with its dual space via the given scalar product, we interpret a symmetric tensor of rank p as a symmetric p-linear map from (R n ) p to R. A special tensor is the metric tensor Q ∈ T 2 , defined by Q(x, y) := x, y for x, y ∈ R n .…”

Section: Preliminariesmentioning

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: Preliminariesmentioning

confidence: 99%

Kinematic formulae for tensorial curvature measures

Hug

Weis

2018

Annali di Matematica

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“…Theorem 24 (Wannerer [47]). Let A be the local additive kinematic operator, and a r 1 ,r 2 the additive kinematic operator for tensor valuations (see [21,22] From the theorem we can derive a strategy to compute the operator A, i.e. the local additive kinematic formulas: we have to choose r 1 , r 2 in such a way that M r 1 and M r 2 are injective, and then a r 1 ,r 2 will determine A.…”

Section: The Moment Map and Additive Kinematic Formulas For Tensor Vamentioning

confidence: 99%

“…He then went on to compute the relevant parts of the additive kinematic formula a 2,2 for unitarily invariant tensor valuations. He first proved an additive version of Theorem 10 for tensor valuations, relating a and the convolution product of tensor valuations (compare also [21]). The convolution product of unitarily invariant tensor valuations of rank 2 can be computed using a formula from [18].…”

Section: Hermitian Casementioning

confidence: 99%

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Valuations and Curvature Measures on Complex Spaces

Bernig

2017

Lecture Notes in Mathematics

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We survey recent results in hermitian integral geometry, i.e. integral geometry on complex vector spaces and complex space forms. We study valuations and curvature measures on complex space forms and describe how the global and local kinematic formulas on such spaces were recently obtained. While the local and global kinematic formulas in the Euclidean case are formally identical, the local formulas in the hermitian case contain strictly more information than the global ones. Even if one is only interested in the flat hermitian case, i.e. C n , it is necessary to study the family of all complex space forms, indexed by the holomorphic curvature 4λ , and the dependence of the formulas on the parameter λ . We will also describe Wannerer's recent proof of local additive kinematic formulas for unitarily invariant area measures.

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“…An extensive theory of integral geometric intersection formulas for tensor valued valuations, also with values in spaces of measures, was developed in [2,5,7,8,9,13,15]. We also refer to the survey article [1].…”

Section: Introductionmentioning

confidence: 99%

Cauchy’s surface area formula in the Heisenberg groups

Huang

2021

Rev. Mat. Iberoam.

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Cauchy's surface area formula says that for a convex body K in n-dimensional Euclidean space the mean value of the (n − 1)-dimensional volumes of the orthogonal projections of K to hyperplanes is a constant multiple of the surface area of K. We prove an analogous formula, with the volumes of the projections replaced by their moment vectors. This requires to introduce a new vector-valued valuation on convex bodies.

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Integral Geometry and Algebraic Structures for Tensor Valuations

Cited by 5 publications

References 12 publications

Kinematic formulae for tensorial curvature measures

Kinematic formulae for tensorial curvature measures

Valuations and Curvature Measures on Complex Spaces

Cauchy’s surface area formula in the Heisenberg groups

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