2008
DOI: 10.1016/j.aam.2008.04.001
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Integral geometry of tensor valuations

Abstract: We prove a complete set of integral geometric formulas of Crofton type (involving integrations over affine Grassmannians) for the Minkowski tensors of convex bodies. Minkowski tensors are the natural tensor valued valuations generalizing the intrinsic volumes (or Minkowski functionals) of convex bodies. By Hadwiger's general integral geometric theorem, the Crofton formulas yield also kinematic formulas for Minkowski tensors. The explicit calculations of integrals over affine Grassmannians require several integ… Show more

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Cited by 63 publications
(86 citation statements)
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“…The tensor Cm is equal to a linear combination of the Minkowski tensor W10,2 (which is defined below, see Tables I and II) and the unit tensor multiplied by the surface area (or the perimeter for d = 2). The coefficients are explicitly given by so‐called Crofton formulas for Minkowski tensors . For local versions of these Crofton formulas, see Ref.…”
Section: Mean Intercept Length Of Anisotropic Boolean Modelsmentioning
confidence: 99%
See 1 more Smart Citation
“…The tensor Cm is equal to a linear combination of the Minkowski tensor W10,2 (which is defined below, see Tables I and II) and the unit tensor multiplied by the surface area (or the perimeter for d = 2). The coefficients are explicitly given by so‐called Crofton formulas for Minkowski tensors . For local versions of these Crofton formulas, see Ref.…”
Section: Mean Intercept Length Of Anisotropic Boolean Modelsmentioning
confidence: 99%
“…The coefficients are explicitly given by so-called Crofton formulas for Minkowski tensors. 41,42 For local versions of these Crofton formulas, see Ref. [43].…”
Section: F Generalized Anisotropy Measuresmentioning
confidence: 99%
“…Now we shall derive a local Steiner formula and support measures, following [23] to obtain local support measures. The definition of MF and MT then follows directly as a special case [33,62]. The intuitive idea underlying the definition of a local parallel set is described in two steps.…”
Section: Definition Based On Fundamental Measure Theorymentioning
confidence: 99%
“…Some other important particular cases of valuations are given, for instance, when considering the vector space of symmetric tensors (see [2,9,20,30,37] for more information on tensor-valued valuations), or (K(V ), +) with + the Minkowski sum between two convex bodies (i.e. K + L = {x + y: x ∈ K, y ∈ L}).…”
Section: Introductionmentioning
confidence: 99%