Geometric measures for random mosaics in spherical spaces

Arbeiter, E.; Zähle, M.

doi:10.1080/17442509408833870

Cited by 18 publications

(37 citation statements)

References 10 publications

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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%

“…On the other hand, in recent years non-Euclidean models in stochastic geometry have attracted increasing attention, in particular in random geometric structures on manifolds, see e.g. [3,5,6,12,17,18,19,28,29,30,31,41,42]. 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.…”

Section: Introductionmentioning

confidence: 99%

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Splitting tessellations in spherical spaces

Hug¹,

Thäle²

2019

Electron. J. Probab.

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The concept of splitting tessellations and splitting tessellation processes in spherical spaces of dimension d ≥ 2 is introduced. Expectations, variances and covariances of spherical curvature measures induced by a splitting tessellation are studied using tools from spherical integral geometry. Also the spherical pair-correlation function of the (d − 1)-dimensional Hausdorff measure is computed explicitly and compared to its analogue for Poisson great hypersphere tessellations. Finally, the typical cell distribution and the distribution of the typical spherical maximal face of any dimension k ∈ {1, . . . , d − 1} are expressed as mixtures of the related distributions of Poisson great hypersphere tessellations. This in turn is used to determine the expected length and the precise birth time distribution of the typical maximal spherical segment of a splitting tessellation. From now on we call elements of S d−1 great hyperspheres of S d . For a set B ⊂ S d , let us denote by S d−1 [B] := {S ∈ S d−1 : S ∩ int(B) = ∅} the set of all great hyperspheres of S d which have non-empty intersection with the interior int(B) of B (here the interior refers to the topology of S d ). Definition 2.1. A Borel measure κ on S d−1 is called regular if κ({S ∈ S d−1 : e ∈ S}) = 0 for e ∈ S d .Clearly, ν d−1 and any measure which is absolutely continuous with respect to ν d−1 is regular. The assertion of the following lemma will be used repeatedly. For S ∈ S d−1 we denote by S + , S − the two closed half-spheres bounded by S. Since we shall always work with statements symmetric in S + , S − , we do not have to specify this further. Lemma 2.2. If κ is a regular Borel measure on S d−1 and c ∈ P d , then κ({S ∈ S d−1 : S ∩ c = ∅, c ⊂ S + or c ⊂ S − }) = 0 .This remains true for an arbitrary compact set C ⊂ S d in place of c if κ is absolutely continuous with respect to ν d−1 .

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Section: Typical Maximal Spherical Facesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Splitting tessellations in spherical spaces

Hug¹,

Thäle²

2019

Electron. J. Probab.

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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%

A random cell splitting scheme on the sphere

Deuß

Hörrmann

Thäle

2017

Stochastic Processes and their Applications

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Abstract. A random recursive cell splitting scheme of the 2-dimensional unit sphere is considered, which is the spherical analogue of the STIT tessellation process from Euclidean stochastic geometry. First-order moments are computed for a large array of combinatorial and metric parameters of the induced splitting tessellations by means of martingale methods combined with tools from spherical integral geometry. The findings are compared with those in the Euclidean case, making thereby transparent the influence of the curvature of the underlying space. Moreover, the capacity functional is computed and the point process that arises from the intersection of a splitting tessellation with a fixed great circle is characterized.

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Tessellations

Schoenberg

2006

Encyclopedia of Environmetrics

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A tessellation may be defined as a division of a space into convex polygonal regions; divisions of the plane ( R 2 ) are most often discussed. While geometrists centuries ago reserved the term tessellation merely for divisions of the plane into regular polygons of equal size, most authors now define tessellations very broadly so as to include any arrangements of nonoverlapping shapes covering R d , as well as partitions of other metric spaces.

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Geometric measures for random mosaics in spherical spaces

Cited by 18 publications

References 10 publications

Splitting tessellations in spherical spaces

Splitting tessellations in spherical spaces

A random cell splitting scheme on the sphere

Tessellations

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