Intrinsic Volumes of the Maximal Polytope Process in Higher Dimensional STIT Tessellations

Stationary and isotropic iteration stable random tessellations are considered, which can be constructed by a random process of cell division. The collection of maximal polytopes at a fixed time $t$ within a convex window $W\subset{\Bbb R}^d$ is regarded and formulas for mean values, variances, as well as a characterization of certain covariance measures are proved. The focus is on the case $d\geq 3$, which is different from the planar one, treated separately in \cite{ST2}. Moreover, a multivariate limit theorem for the vector of suitably rescaled intrinsic volumes is established, leading in each component -- in sharp contrast to the situation in the plane -- to a non-Gaussian limit.Comment: 27 page

Counting faces of randomly-projected polytopes when the projection radically lowers dimension

This paper develops asymptotic methods to count faces of random high-dimensional polytopes. Beyond its intrinsic interest, our conclusions have surprising implications - in statistics, probability, information theory, and signal processing - with potential impacts in practical subjects like medical imaging and digital communications. Three such implications concern: convex hulls of Gaussian point clouds, signal recovery from random projections, and how many gross errors can be efficiently corrected from Gaussian error correcting codes.Comment: 56 page

Random Inscribed Polytopes Have Similar Radius Functions as Poisson-Delaunay Mosaics

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