Expected Sizes of Poisson–Delaunay Mosaics and Their Discrete Morse Functions∗

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, we study the expected number of simplices in the Delaunay mosaic as well as the expected number of critical simplices and non-singular 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 get the expected numbers of simplices in the Poisson–Delaunay mosaic in dimensions n ≤ 4. 1998 ACM Subject Classification: I.3.5 Computational Geometry and Object Modeling, G.3 Probability and Statistics, G.2 Discrete Mathematics 2010 AMS Mathematics Subject Classification: 60D05 Geometric probability and stochastic geometry, 68U05 Computer graphics; computational geometry