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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2508.05194 |
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Table of Contents:
- We investigate how many hyperplanes with independent standard Gaussian directions one needs to produce a $δ$-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 $δ$. It was conjectured that $δ^{-2}w_*(S)^2$ 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 $δ^{-3}w_*(S)^2$ Gaussian hyperplanes are necessary and sufficient.