Local extrema for hypercube sections
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866929394634719232 |
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| author | Pournin, Lionel |
| author_facet | Pournin, Lionel |
| contents | Consider the hyperplanes at a fixed distance $t$ from the center of the hypercube $[0,1]^d$. Significant attention has been given to determining the hyperplanes $H$ among these such that the $(d-1)$-dimensional volume of $H\cap[0,1]^d$ is maximal or minimal. In the spirit of a question by Vitali Milman, the corresponding local problem is considered here when $H$ is orthogonal to a diagonal or a sub-diagonal of the hypercube. It is proven in particular that this volume is strictly locally maximal at the diagonals in all dimensions greater than $3$ within a range for $t$ that is asymptotic to $\sqrt{d}/\!\log d$. At lower order sub-diagonals, this volume is shown to be strictly locally maximal when $t$ is close to $0$ and not locally extremal when $t$ is large. This relies on a characterisation of local extremality at the diagonals and sub-diagonals that allows to solve the problem over the whole possible range for $t$ in any fixed, reasonably low dimension. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_15054 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Local extrema for hypercube sections Pournin, Lionel Metric Geometry Functional Analysis Consider the hyperplanes at a fixed distance $t$ from the center of the hypercube $[0,1]^d$. Significant attention has been given to determining the hyperplanes $H$ among these such that the $(d-1)$-dimensional volume of $H\cap[0,1]^d$ is maximal or minimal. In the spirit of a question by Vitali Milman, the corresponding local problem is considered here when $H$ is orthogonal to a diagonal or a sub-diagonal of the hypercube. It is proven in particular that this volume is strictly locally maximal at the diagonals in all dimensions greater than $3$ within a range for $t$ that is asymptotic to $\sqrt{d}/\!\log d$. At lower order sub-diagonals, this volume is shown to be strictly locally maximal when $t$ is close to $0$ and not locally extremal when $t$ is large. This relies on a characterisation of local extremality at the diagonals and sub-diagonals that allows to solve the problem over the whole possible range for $t$ in any fixed, reasonably low dimension. |
| title | Local extrema for hypercube sections |
| topic | Metric Geometry Functional Analysis |
| url | https://arxiv.org/abs/2203.15054 |