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 ℝn, we study the expected number of simplices in the Delaunay mosaic as well as the expected number of critical simplices and nonsingular 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 obtain the expected numbers of simplices in the Poisson–Delaunay mosaic in dimensions n ≤ 4.

We show that the expected number of maximal empty axis-parallel boxes amidst n random points in the unit hypercube [0,1]d in $\mathbb{R}^d$ is (1 ± o(1)) $\frac{(2d-2)!}{(d-1)!}$n lnd−1n, if d is fixed. This estimate is relevant to analysis of the performance of exact algorithms for computing the largest empty axis-parallel box amidst n given points in an axis-parallel box R, especially the algorithms that proceed by examining all maximal empty boxes. Our method for bounding the expected number of maximal empty boxes also shows that the expected number of maximal empty orthants determined by n random points in $\mathbb{R}^d$ is (1 ± o(1)) lnd−1n, if d is fixed. This estimate is related to the expected number of maximal (or minimal) points amidst random points, and has application to algorithms for coloured orthogonal range counting.