Sharp estimates on random hyperplane tessellations

TL;DR

This paper establishes precise bounds on the number of hyperplanes needed for random Gaussian tessellations to accurately represent Euclidean distances in any set, challenging previous conjectures about optimality.

Contribution

It provides sharp bounds on hyperplane counts for random tessellations, disproving the conjecture that the Gaussian width squared over delta squared is optimal.

Findings

Lower bounds contradict the conjecture on optimal hyperplane count

Derived sharp bounds for Gaussian tessellations

Showed that the number of hyperplanes can be smaller than previously believed

Abstract

We study the problem of generating a hyperplane tessellation of an arbitrary set $T$ in $\mathbb{R}^n$, ensuring that the Euclidean distance between any two points corresponds to the fraction of hyperplanes separating them up to a pre-specified error $\delta$. We focus on random gaussian tessellations with uniformly distributed shifts and derive sharp bounds on the number of hyperplanes $m$ that are required. Surprisingly, our lower estimates falsify the conjecture that $m\sim \ell_*^2(T)/\delta^2$, where $\ell_*^2(T)$ is the gaussian width of $T$, is optimal.