A limit theorem for Hausdorff approximation by random inscribed polytopes
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866916912176300032 |
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| author | Sonnleitner, Mathias |
| author_facet | Sonnleitner, Mathias |
| contents | Approximate a smooth convex body $K$ with nonvanishing curvature by the convex hull of $n$ independent random points sampled from its boundary $\partial K$. In case the points are distributed according to the optimal density, we prove that the rescaled approximation error in Hausdorff distance tends to a Gumbel distributed random variable. The proof is based on an asymptotic relation to covering properties of random geodesic balls on $\partial K$ and on a limit theorem due to Janson. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_16442 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A limit theorem for Hausdorff approximation by random inscribed polytopes Sonnleitner, Mathias Probability Metric Geometry 52A27, 60D05, 60F05 (Primary) 52C17, 60G70 (Secondary) Approximate a smooth convex body $K$ with nonvanishing curvature by the convex hull of $n$ independent random points sampled from its boundary $\partial K$. In case the points are distributed according to the optimal density, we prove that the rescaled approximation error in Hausdorff distance tends to a Gumbel distributed random variable. The proof is based on an asymptotic relation to covering properties of random geodesic balls on $\partial K$ and on a limit theorem due to Janson. |
| title | A limit theorem for Hausdorff approximation by random inscribed polytopes |
| topic | Probability Metric Geometry 52A27, 60D05, 60F05 (Primary) 52C17, 60G70 (Secondary) |
| url | https://arxiv.org/abs/2508.16442 |