The expected number of facets and the typical facet height of the convex hull of n uniform points on S^{d-1} are determined asymptotically in every regime where n and d tend to infinity.
The stochastic geometry of unconstrained one-bit data compression
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abstract
A stationary stochastic geometric model is proposed for analyzing the data compression method used in one-bit compressed sensing. The data set is an unconstrained stationary set, for instance all of $\mathbb{R}^n$ or a stationary Poisson point process in $\mathbb{R}^n$. It is compressed using a stationary and isotropic Poisson hyperplane tessellation, assumed independent of the data. That is, each data point is compressed using one bit with respect to each hyperplane, which is the side of the hyperplane it lies on. This model allows one to determine how the intensity of the hyperplanes must scale with the dimension $n$ to ensure sufficient separation of different data by the hyperplanes as well as sufficient proximity of the data compressed together. The results have direct implications in compressive sensing and in source coding.
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Facets of spherical random polytopes
The expected number of facets and the typical facet height of the convex hull of n uniform points on S^{d-1} are determined asymptotically in every regime where n and d tend to infinity.