A resolution of the Gaussian hyperplane tessellation conjecture on the sphere
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | |
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| _version_ | 1866913978890846208 |
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| author | Dirksen, Sjoerd Strachan, Nigel Q. D. |
| author_facet | Dirksen, Sjoerd Strachan, Nigel Q. D. |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_05194 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A resolution of the Gaussian hyperplane tessellation conjecture on the sphere Dirksen, Sjoerd Strachan, Nigel Q. D. Probability 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. |
| title | A resolution of the Gaussian hyperplane tessellation conjecture on the sphere |
| topic | Probability |
| url | https://arxiv.org/abs/2508.05194 |