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Suppose we choose $N$ points uniformly randomly from a convex body in $d$ dimensions. How large must $N$ be, asymptotically with respect to $d$, so that the convex hull of the points is nearly as large as the convex body itself? It was shown by Dyer-Furedi-McDiarmid that exponentially many samples suffice when the convex body is the hypercube, and by Pivovarov that the Euclidean ball demands roughly $d^{d/2}$ samples. We show that when the convex body is the simplex, exponentially many samples suffice; this then implies the same result for any convex simplicial polytope with at most exponentially many faces.
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A new approach to prove weak convergence of random polytopes on the space of compact convex sets is presented. This is used to show that the profile of the rescaled Schlafli random cone of a random conical tessellation generated by $n$ independent and uniformly distributed random linear hyperplanes in $mathbb{R}^{d+1}$ weakly converges to the typical cell of a stationary and isotropic Poisson hyperplane tessellation in $mathbb{R}^d$, as $n to infty$.
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.
A two-step model for generating random polytopes is considered. For parameters $d$, $m$, and $p$, the first step is to generate a simple polytope $P$ whose facets are given by $m$ uniform random hyperplanes tangent to the unit sphere in $mathbb{R}^d$, and the second step is to sample each vertex of $P$ independently with probability $p$ and let $Q$ be the convex hull of the sampled vertices. We establish results on how well $Q$ approximates the unit sphere in terms of $m$ and $p$ as well as asymptotics on the combinatorial complexity of $Q$ for certain regimes of $p$.
We propose a new geometric method for measuring the quality of representations obtained from deep learning. Our approach, called Random Polytope Descriptor, provides an efficient description of data points based on the construction of random convex polytopes. We demonstrate the use of our technique by qualitatively comparing the behavior of classic and regularized autoencoders. This reveals that applying regularization to autoencoder networks may decrease the out-of-distribution detection performance in latent space. While our technique is similar in spirit to $k$-means clustering, we achieve significantly better false positive/negative balance in clustering tasks on autoencoded datasets.
The random convex hull of a Poisson point process in $mathbb{R}^d$ whose intensity measure is a multiple of the standard Gaussian measure on $mathbb{R}^d$ is investigated. The purpose of this paper is to invent a new viewpoint on these Gaussian polytopes that is based on cumulants and the general large deviation theory of Saulis and Statuleviv{c}ius. This leads to new and powerful concentration inequalities, moment bounds, Marcinkiewicz-Zygmund-type strong laws of large numbers, central limit theorems and moderate deviation principles for the volume and the face numbers. Corresponding results are also derived for the empirical measures induced by these key geometric functionals, taking thereby care of their spatial profiles.
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