This article is cited in 1 paper

Weighted Poisson–Delaunay mosaics

H. Edelsbrunner, A. Nikitenko

Institute of Science and Technology Austria, Klosterneuburg, Austria

Abstract: Slicing a Voronoi tessellation in $\mathbf{R}^n$ with a $k$-plane gives a $k$-dimensional weighted Voronoi tessellation, also known as a 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 $\mathbf{R}^n$, 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 by-product, we obtain a new proof for the expected number of connected components (clumps) in a line section of a circular Boolean model in $\mathbf{R}^n$.

Keywords: Voronoi tessellations, Laguerre distance, weighted Delaunay mosaics, discrete Morse theory, critical simplices, intervals, stochastic geometry, Poisson point process, Boolean model, clumps, Slivnyak–Mecke formula, Blaschke–Petkantschin formula.

MSC: 60D05; 68U05

Received: 17.03.2018

DOI: 10.4213/tvp5196

 English version: Theory of Probability and its Applications, 2020, 64:4, 595–614

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