Approximate isoperimetry for convex polytopes
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909792847527936 |
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| author | Ball, Keith Böröczky, Károly J. Naor, Assaf |
| author_facet | Ball, Keith Böröczky, Károly J. Naor, Assaf |
| contents | For all $n,ϕ\in \mathbb{N}$ with $ϕ\geqslant n+1$, the smallest possible isoperimetric quotient of an $n$-dimensional convex polytope that has $ϕ$ facets is shown to be bounded from above and from below by positive universal constant multiples of $\max\big\{n/\sqrt{1+\log (ϕ/n)},\sqrt{n}\big\}$. For all $n\in \mathbb{N}$ and $2n\leqslant β\in 2\mathbb{N}$, it is shown that every $n$-dimensional origin-symmetric convex polytope that has $β$ vertices admits an affine image whose isoperimetric quotient is at most a universal constant multiple of $\min\big\{\sqrt{\log(β/n)},n\big\}$, which is sharp. The weak isomorphic reverse isoperimetry conjecture is proved for $n$-dimensional convex polytopes that have $O(n)$ facets by demonstrating that any such polytope $K$ has an image $K'$ under a volume preserving matrix and a convex body $L\subseteq K'$ such that the isoperimetric quotient of $L$ is at most a universal constant multiple of $\sqrt{n}$, and also $\sqrt[n]{\mathrm{vol}_n(L)/\mathrm{vol}_n(K)}$ is at least a positive universal constant. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_13898 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Approximate isoperimetry for convex polytopes Ball, Keith Böröczky, Károly J. Naor, Assaf Metric Geometry Functional Analysis For all $n,ϕ\in \mathbb{N}$ with $ϕ\geqslant n+1$, the smallest possible isoperimetric quotient of an $n$-dimensional convex polytope that has $ϕ$ facets is shown to be bounded from above and from below by positive universal constant multiples of $\max\big\{n/\sqrt{1+\log (ϕ/n)},\sqrt{n}\big\}$. For all $n\in \mathbb{N}$ and $2n\leqslant β\in 2\mathbb{N}$, it is shown that every $n$-dimensional origin-symmetric convex polytope that has $β$ vertices admits an affine image whose isoperimetric quotient is at most a universal constant multiple of $\min\big\{\sqrt{\log(β/n)},n\big\}$, which is sharp. The weak isomorphic reverse isoperimetry conjecture is proved for $n$-dimensional convex polytopes that have $O(n)$ facets by demonstrating that any such polytope $K$ has an image $K'$ under a volume preserving matrix and a convex body $L\subseteq K'$ such that the isoperimetric quotient of $L$ is at most a universal constant multiple of $\sqrt{n}$, and also $\sqrt[n]{\mathrm{vol}_n(L)/\mathrm{vol}_n(K)}$ is at least a positive universal constant. |
| title | Approximate isoperimetry for convex polytopes |
| topic | Metric Geometry Functional Analysis |
| url | https://arxiv.org/abs/2509.13898 |