EXPECTED SIZES OF POISSON-DELAUNAY MOSAICS AND THEIR DISCRETE MORSE FUNCTIONS

Autor(en): Edelsbrunner, Herbert
Nikitenko, Anton
Reitzner, Matthias 
Stichwörter: critical simplices; Delaunay mosaic; discrete Morse theory; HOMOLOGY; integral geometry; intervals; Mathematics; Poisson point process; Statistics & Probability; stochastic geometry; TESSELLATIONS; TOPOLOGY; typical simplex
Erscheinungsdatum: 2017
Herausgeber: APPLIED PROBABILITY TRUST
Journal: ADVANCES IN APPLIED PROBABILITY
Volumen: 49
Ausgabe: 3
Startseite: 745
Seitenende: 767
Zusammenfassung: 
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.
ISSN: 00018678
DOI: 10.1017/apr.2017.20

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