Sharp estimates on random hyperplane tessellations

01/13/2022

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by   Sjoerd Dirksen, et al.

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We study the problem of generating a hyperplane tessellation of an arbitrary set T in 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 δ. 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∼ℓ_*^2(T)/δ^2, where ℓ_*^2(T) is the gaussian width of T, is optimal.