Expected sizes of Poisson–Delaunay mosaics and their discrete Morse functions. (September 2017)

• Record Type: Journal Article
• Authors: Edelsbrunner, Herbert; Nikitenko, Anton; Reitzner, Matthias
• Is Part Of: Advances in applied probability. Volume 49:Number 3(2017)
• Page Start: 745
• Page End: 767
• Publication Date: 2017-09
• Subjects: Poisson point process, -- Delaunay mosaic, -- discrete Morse theory, -- critical simplices, -- intervals, -- stochastic geometry, -- integral geometry, -- typical simplex; Primary 60D05, -- Secondary 68U05; Probabilities -- Periodicals; Stochastic models -- Periodicals; Electronic journals; Periodicals; 519.2
• DOI: 10.1017/apr.2017.20
• Languages: English
• ISSNs: 0001-8678

Abstract

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.