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

  • Herbert Edelsbrunner (a1), Anton Nikitenko (a1) and Matthias Reitzner (a2)

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 ℝ n , 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.

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Corresponding author

* Postal address: IST Austria, Am Campus 1, 3400 Klosterneuburg, Austria.
** Email address: edels@ist.ac.at
*** Email address: anton.nikitenko@ist.ac.at
**** Postal address: Mathematics Department, University of Osnabrück, Albrechtstrasse 28a, 49076 Osnabrück, Germany. Email address: mreitzne@uni-osnabrueck.de

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

  • Herbert Edelsbrunner (a1), Anton Nikitenko (a1) and Matthias Reitzner (a2)

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