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Expected sizes of Poisson–Delaunay mosaics and their discrete Morse functions

Part of: Geometric probability and stochastic geometry

Published online by Cambridge University Press:  08 September 2017

Herbert Edelsbrunner ,
Anton Nikitenko  and
Matthias Reitzner
Herbert Edelsbrunner*
Affiliation:
IST Austria
Anton Nikitenko*
Affiliation:
IST Austria
Matthias Reitzner*
Affiliation:
University of Osnabrück
*
* Postal address: IST Austria, Am Campus 1, 3400 Klosterneuburg, Austria.
* Postal address: IST Austria, Am Campus 1, 3400 Klosterneuburg, Austria.
**** Postal address: Mathematics Department, University of Osnabrück, Albrechtstrasse 28a, 49076 Osnabrück, Germany. Email address: mreitzne@uni-osnabrueck.de
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Abstract

Mapping every simplex in the Delaunay mosaic of a discrete point set to the radius of the smallest empty circumsphere gives a generalized discrete Morse function. Choosing the points from a Poisson point process in ℝn, we study the expected number of simplices in the Delaunay mosaic as well as the expected number of critical simplices and nonsingular intervals in the corresponding generalized discrete gradient. Observing connections with other probabilistic models, we obtain precise expressions for the expected numbers in low dimensions. In particular, we obtain the expected numbers of simplices in the Poisson–Delaunay mosaic in dimensions n ≤ 4.

Keywords

Poisson point processDelaunay mosaicdiscrete Morse theorycritical simplicesintervalsstochastic geometryintegral geometrytypical simplex

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 68U05: Computer graphics; computational geometry
Type
Research Article
Information
Advances in Applied Probability , Volume 49 , Issue 3 , September 2017 , pp. 745 - 767
Copyright
Copyright © Applied Probability Trust 2017 

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