Polynomial Bounds in Koldobsky's Discrete Slicing Problem
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866929222720684032 |
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| author | Freyer, Ansgar Henk, Martin |
| author_facet | Freyer, Ansgar Henk, Martin |
| contents | In 2013, Koldobsky posed the problem to find a constant $d_n$, depending only on the dimension $n$, such that for any origin-symmetric convex body $K\subset\mathbb{R}^n$ there exists an $(n-1)$-dimensional linear subspace $H\subset\mathbb{R}^n$ with \[
|K\cap\mathbb Z^n| \leq d_n\,|K\cap H\cap \mathbb Z^n|\,\mathrm{vol}(K)^{\frac 1n}. \] In this article we show that $d_n$ is bounded from above by $c\,n^2\,ω(n)/\log(n)$, where $c$ is an absolute constant and $ω(n)$ is the flatness constant. Due to the recent best known upper bound on $ω(n)$ we get a ${c\,n^3\log(n)^2}$ bound on $d_n$. This improves on former bounds which were exponential in the dimension. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_15976 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Polynomial Bounds in Koldobsky's Discrete Slicing Problem Freyer, Ansgar Henk, Martin Metric Geometry In 2013, Koldobsky posed the problem to find a constant $d_n$, depending only on the dimension $n$, such that for any origin-symmetric convex body $K\subset\mathbb{R}^n$ there exists an $(n-1)$-dimensional linear subspace $H\subset\mathbb{R}^n$ with \[ |K\cap\mathbb Z^n| \leq d_n\,|K\cap H\cap \mathbb Z^n|\,\mathrm{vol}(K)^{\frac 1n}. \] In this article we show that $d_n$ is bounded from above by $c\,n^2\,ω(n)/\log(n)$, where $c$ is an absolute constant and $ω(n)$ is the flatness constant. Due to the recent best known upper bound on $ω(n)$ we get a ${c\,n^3\log(n)^2}$ bound on $d_n$. This improves on former bounds which were exponential in the dimension. |
| title | Polynomial Bounds in Koldobsky's Discrete Slicing Problem |
| topic | Metric Geometry |
| url | https://arxiv.org/abs/2303.15976 |