On the Number of Faces and Radii of Cells Induced by Gaussian Spherical Tessellations

We study a geometric property related to spherical hyperplane tessellations in $\mathbb{R}^{d}$. We first consider a fixed $x$ on the Euclidean sphere and tessellations with $M \gg d$ hyperplanes passing through the origin having normal vectors distributed according to a Gaussian distribution. We show that with high probability there exists a subset of the hyperplanes whose cardinality is on the order of $d\log(d)\log(M)$ such that the radius of the cell containing $x$ induced by these hyperplanes is bounded above by, up to constants, $d\log(d)\log(M)/M$. We extend this result to hold for all cells in the tessellation with high probability. Up to logarithmic terms, this upper bound matches the previously established lower bound of Goyal et al. (IEEE T. Inform. Theory 44(1):16-31, 1998).