Sharp estimates on random hyperplane tessellations

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.