Electronic Journal of Probability

Splitting tessellations in spherical spaces

Daniel Hug and Christoph Thäle

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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.

Article information

Electron. J. Probab., Volume 24 (2019), paper no. 24, 60 pp.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 52A22: Random convex sets and integral geometry [See also 53C65, 60D05] 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 53C65: Integral geometry [See also 52A22, 60D05]; differential forms, currents, etc. [See mainly 58Axx]

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

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Hug, Daniel; Thäle, Christoph. Splitting tessellations in spherical spaces. Electron. J. Probab. 24 (2019), paper no. 24, 60 pp. doi:10.1214/19-EJP267. https://projecteuclid.org/euclid.ejp/1553565775

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