A resolution of the Gaussian hyperplane tessellation conjecture on the sphere

Abstract: We investigate how many hyperplanes with independent standard Gaussian directions one needs to produce a $\delta$-uniform tessellation of a subset $S$ of the Euclidean sphere, meaning that for any pair of points in $S$ the fraction of hyperplanes separating them corresponds to their geodesic distance up to an additive error $\delta$. It was conjectured that $\delta^{-2}w_*(S)^2$ Gaussian random hyperplanes are necessary and sufficient for this purpose, where $w_(S)$ is the Gaussian complexity of $S$. We falsify this conjecture by constructing a set $S$ where $\delta^{-3}w_(S)^2$ Gaussian hyperplanes are necessary and sufficient.