Weighted Poisson–Delaunay Mosaics∗

Slicing a Voronoi tessellation in R with a k-plane gives a k-dimensional weighted Voronoi tessellation, also known as power diagram or Laguerre tessellation. Mapping every simplex of the dual weighted Delaunay mosaic to the radius of the smallest empty circumscribed sphere whose center lies in the k-plane gives a generalized discrete Morse function. Assuming the Voronoi tessellation is generated by a Poisson point process in R, we study the expected number of simplices in the k-dimensional weighted Delaunay mosaic as well as the expected number of intervals of the Morse function, both as functions of a radius threshold. As a byproduct we obtain a new proof for the expected number of connected components (clumps) in a line section of a circular Boolean model in R. 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.