Spatial Stit Tessellations: Distributional Results for I-Segments

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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A random cell splitting scheme on the sphere

Deuß

Hörrmann

Thäle

2017

Stochastic Processes and their Applications

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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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Spatial Stit Tessellations: Distributional Results for I-Segments

Cited by 2 publications

References 22 publications

A random cell splitting scheme on the sphere

A random cell splitting scheme on the sphere

Splitting tessellations in spherical spaces

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