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Slicing a Voronoi tessellation in $mathbb{R}^n$ 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 $mathbb{R}^n$, 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 byproduct, we obtain a new proof for the expected number of connected components (clumps) in a line section of a circular Boolean model in $mathbb{R}^n$
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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 an n-dimensional Poisson point process, we study the expected number of simplices in the Delaunay mosaic as well as the expected number of critical simplices and non-singular 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 get the expected numbers of simplices in the Poisson-Delaunay mosaic in dimensions up to 4.
Using the geodesic distance on the $n$-dimensional sphere, we study the expected radius function of the Delaunay mosaic of a random set of points. Specifically, we consider the partition of the mosaic into intervals of the radius function and determine the expected number of intervals whose radii are less than or equal to a given threshold. Assuming the points are not contained in a hemisphere, the Delaunay mosaic is isomorphic to the boundary complex of the convex hull in $mathbb{R}^{n+1}$, so we also get the expected number of faces of a random inscribed polytope. We find that the expectations are essentially the same as for the Poisson-Delaunay mosaic in $n$-dimensional Euclidean space. As proved by Antonelli and collaborators, an orthant section of the $n$-sphere is isometric to the standard $n$-simplex equipped with the Fisher information metric. It follows that the latter space has similar stochastic properties as the $n$-dimensional Euclidean space. Our results are therefore relevant in information geometry and in population genetics.
The order-$k$ Voronoi tessellation of a locally finite set $X subseteq mathbb{R}^n$ decomposes $mathbb{R}^n$ 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.
The convex hull generated by the restriction to the unit ball of a stationary Poisson point process in the $d$-dimensional Euclidean space is considered. By establishing sharp bounds on cumulants, exponential estimates for large deviation probabilities are derived and the relative error in the central limit theorem on a logarithmic scale is investigated for a large class of key geometric characteristics. This includes the number of lower-dimensional faces and the intrinsic volumes of the random polytopes. Furthermore, moderate deviation principles for the spatial empirical measures induced by these functionals are also established using the method of cumulants. The results are applied to deduce, by duality, fine probabilistic estimates and moderate deviation principles for combinatorial parameters of a class of zero cells associated with Poisson hyperplane mosaics. As a special case this comprises the typical Poisson-Voronoi cell conditioned on having large inradius.
Let $Z$ be the typical cell of a stationary Poisson hyperplane tessellation in $mathbb{R}^d$. The distribution of the number of facets $f(Z)$ of the typical cell is investigated. It is shown, that under a well-spread condition on the directional distribution, the quantity $n^{frac{2}{d-1}}sqrt[n]{mathbb{P}(f(Z)=n)}$ is bounded from above and from below. When $f(Z)$ is large, the isoperimetric ratio of $Z$ is bounded away from zero with high probability. These results rely on one hand on the Complementary Theorem which provides a precise decomposition of the distribution of $Z$ and on the other hand on several geometric estimates related to the approximation of polytopes by polytopes with fewer facets. From the asymptotics of the distribution of $f(Z)$, tail estimates for the so-called $Phi$ content of $Z$ are derived as well as results on the conditional distribution of $Z$ when its $Phi$ content is large.
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