1994
DOI: 10.1080/17442509408833870
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Geometric measures for random mosaics in spherical spaces

Abstract: This paper deals with isotropic randoni mosaics of Sd-' whose ce!!s are piecewise smooth. As geometric measures concentrated on the i-skeleton (i = 0. . . . , d -I ) the i-dimensional surface area (volume) measure and (i -I ) different curvature measures are chosei;. Relations be!ween the corresponding densities are found. This leads to a parameier systein from which 211 mean values can be calculated. The curvature intensities are alsu compiited for the intersec!im with a subsphere, for superposition and for g… Show more

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Cited by 18 publications
(37 citation statements)
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“…For this we denote by N k (t) = F (k) t (P d k ) the expected number of maximal spherical k-faces of the splitting tessellation Y t and by N k (t) = F (k) t (P d k ) the expected number of spherical k-faces of the Poisson great hypersphere tessellation Y t (as described above). The computation of N k (t) or N k (t) is in general rather involved as considerations in [3,18] indicate. For this reason we shall carry out explicit computations only for the special case k = 1 below.…”
Section: Typical Maximal Spherical Facesmentioning
confidence: 99%
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“…For this we denote by N k (t) = F (k) t (P d k ) the expected number of maximal spherical k-faces of the splitting tessellation Y t and by N k (t) = F (k) t (P d k ) the expected number of spherical k-faces of the Poisson great hypersphere tessellation Y t (as described above). The computation of N k (t) or N k (t) is in general rather involved as considerations in [3,18] indicate. For this reason we shall carry out explicit computations only for the special case k = 1 below.…”
Section: Typical Maximal Spherical Facesmentioning
confidence: 99%
“…More specifically, the papers [6,28,29] deal with spherical convex hulls, [5,19] are concerned with spherical convex hulls of random points on half-spheres, the work [41] considers central limit theorems for point process statistics of point processes on manifolds, [42] is devoted to the study of random systems on Cayley graphs, and [30,31] explores geometric aspects of random fields on the sphere. Further results for tessellations of the d-dimensional unit sphere by great hyperspheres have been obtained in [3] and, more recently, in [17,18], which generalize at the same time some mean value computations for tessellations generated by great circles on the 2dimensional sphere S 2 in [34]. On the other hand, in [8] an analysis of Voronoi tessellations in rather general Riemannian manifolds is initiated.…”
Section: Introductionmentioning
confidence: 99%
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“…Let us finally compare some of the mean values we computed with those for Poisson great circle tessellations on the sphere, see [1,10]. To define the model, let for some γ > 0, η γ be a Poisson point process on S 2 whose intensity measure is given by γ σ 2 .…”
Section: Mean Valuesmentioning
confidence: 99%
“…While random tessellations of Euclidean spaces, especially of the Euclidean plane, are a classical topic in stochastic geometry (see Chapter 10 in [14] and the references given therein), random tessellations on the sphere have not found equal attention in the existing literature. However, random tessellations of S 2 or higher-dimensional spherical spaces by great circles were studied in [1,10,13] and also the recent work [4] on so-called conical random tessellation is closely related to this model. Spherical random Voronoi tessellations and their applications were investigated in [10,18,19,23].…”
Section: Introductionmentioning
confidence: 99%