A resolution of the Gaussian hyperplane tessellation conjecture on the sphere

TL;DR

This paper resolves a conjecture about the number of Gaussian hyperplanes needed for uniform tessellations on the sphere, showing the conjectured bound is not tight and providing a new necessary and sufficient condition.

Contribution

It disproves the existing conjecture by constructing a set requiring more hyperplanes, establishing a new bound involving  w_*(S)^2 hyperplanes.

Findings

Disproved the Gaussian hyperplane tessellation conjecture.

Constructed a set requiring  w_*(S)^2 hyperplanes.

Provided a new bound for hyperplane requirements.

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.