Poisson–Delaunay Mosaics of Order k

The order-k Voronoi tessellation of a locally finite set [Formula: see text] decomposes [Formula: see text] into convex domains whose points have the same k nearest neighbors in X. Assuming X is a stationary Poisson point process, we give explicit formulas for the expected number and total area of faces of a given dimension per unit volume of space. We also develop a relaxed version of discrete Morse theory and generalize by counting only faces, for which the k nearest points in X are within a given distance threshold.