Open Access
2019 Splitting tessellations in spherical spaces
Daniel Hug, Christoph Thäle
Electron. J. Probab. 24: 1-60 (2019). DOI: 10.1214/19-EJP267


The concept of splitting tessellations and splitting tessellation processes in spherical spaces of dimension $d\geq 2$ 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 $k\in \{1,\ldots ,d-1\}$ 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.


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Daniel Hug. Christoph Thäle. "Splitting tessellations in spherical spaces." Electron. J. Probab. 24 1 - 60, 2019.


Received: 26 April 2018; Accepted: 15 January 2019; Published: 2019
First available in Project Euclid: 26 March 2019

zbMATH: 1417.52006
MathSciNet: MR3933203
Digital Object Identifier: 10.1214/19-EJP267

Primary: 52A22 , 60D05
Secondary: 53C65

Keywords: $K$-function , Blaschke-Petkantschin formula , Markov process , martingale , maximal face , pair-correlation function , random tessellation , spherical curvature measure , spherical integral geometry , spherical space , splitting tessellation

Vol.24 • 2019
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