On the $\ell_0$ Isoperimetric Coefficient of Measurable Sets
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866911076247928832 |
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| author | Fernandez V, Manuel |
| author_facet | Fernandez V, Manuel |
| contents | In this paper we prove that the $\ell_0$ isoperimetric coefficient for any axis-aligned cubes, $ψ_{\mathcal{C}}$, is $Θ(n^{-1/2})$ and that the isoperimetric coefficient for any measurable body $K$, $ψ_K$, is of order $O(n^{-1/2})$. As a corollary we deduce that axis-aligned cubes essentially "maximize" the $\ell_0$ isoperimetric coefficient: There exists a positive constant $q > 0$ such that $ψ_K \leq q \cdot ψ_{\mathcal{C}}$, whenever $\mathcal{C}$ is an axis-aligned cube and $K$ is any measurable set. Lastly, we give immediate applications of our results to the mixing time of Coordinate-Hit-and-Run for sampling points uniformly from convex bodies. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_00015 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the $\ell_0$ Isoperimetric Coefficient of Measurable Sets Fernandez V, Manuel Metric Geometry Computational Geometry In this paper we prove that the $\ell_0$ isoperimetric coefficient for any axis-aligned cubes, $ψ_{\mathcal{C}}$, is $Θ(n^{-1/2})$ and that the isoperimetric coefficient for any measurable body $K$, $ψ_K$, is of order $O(n^{-1/2})$. As a corollary we deduce that axis-aligned cubes essentially "maximize" the $\ell_0$ isoperimetric coefficient: There exists a positive constant $q > 0$ such that $ψ_K \leq q \cdot ψ_{\mathcal{C}}$, whenever $\mathcal{C}$ is an axis-aligned cube and $K$ is any measurable set. Lastly, we give immediate applications of our results to the mixing time of Coordinate-Hit-and-Run for sampling points uniformly from convex bodies. |
| title | On the $\ell_0$ Isoperimetric Coefficient of Measurable Sets |
| topic | Metric Geometry Computational Geometry |
| url | https://arxiv.org/abs/2312.00015 |