Author(s):

Edelsbrunner, Herbert; Nikitenko, Anton; Reitzner, Matthias

Article Title: 
Expected sizes of poisson Delaunay mosaics and their discrete Morse functions

Affiliation 
IST Austria 
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.

Keywords: 
intervals; Integral geometry; Poisson point process; Delaunay mosaic; discrete Morse theory; critical simplices; stochastic geometry; typical simplex

Journal Title:

Advances in Applied Probability

Volume: 
49

Issue 
3

ISSN:

14756064

Publisher:

Cambridge University Press

Date Published:

20170101

Start Page: 
745

End Page:

767

URL: 

DOI: 
10.1017/apr.2017.20

Notes: 
This work is partially supported by the Toposys project FP7ICT318493STREP, by ESF
under the ACAT Research Network Programme, and by the FWF within the SFBTransregio
Programme 109 in Discretization in Geometry and Dynamics (grant no. I02979N35).

Open access: 
yes (repository) 