Weighted Poisson--Delaunay mosaics

H. Edelsbrunner ; A. Nikitenko

Teoriâ veroâtnostej i ee primeneniâ, Tome 64 (2019) no. 4, pp. 746-770

Résumé

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 $\mathbf{R}^n$.

Détail

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