On maximal hyperplane sections of the unit ball of $l_p$ for $p>2$
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909466184646656 |
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| author | König, Hermann |
| author_facet | König, Hermann |
| contents | The maximal hyperplane section of the $l_\infty^n$-ball, i.e. of the $n$-cube, is the one perpendicular to 1/sqrt 2 (1,1,0, ... ,0), as shown by Ball. Eskenazis, Nayar and Tkocz extended this result to the $l_p^n$-balls for very large $p \ge 10^{15}$. By Oleszkiewicz, Ball's result does not transfer to $l_p^n$ for $2 < p < p_0 \simeq 26.265$. Then the hyperplane section perpendicular to the main diagonal yields a counterexample for large dimensions $n$. We show that the analogue of Ball's result holds in $l_p^n$-balls for all hyperplanes with normal unit vectors $a$, if all coordinates of $a$ have modulus $\le \frac 1 {\sqrt 2}$ and $p$ has distance $\ge 2^{-p}$ to the even integers. Under similar assumptions, we give a Gaussian upper bound for $20 < p < p_0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_06432 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On maximal hyperplane sections of the unit ball of $l_p$ for $p>2$ König, Hermann Functional Analysis 52A38, 52A40, 46B07 The maximal hyperplane section of the $l_\infty^n$-ball, i.e. of the $n$-cube, is the one perpendicular to 1/sqrt 2 (1,1,0, ... ,0), as shown by Ball. Eskenazis, Nayar and Tkocz extended this result to the $l_p^n$-balls for very large $p \ge 10^{15}$. By Oleszkiewicz, Ball's result does not transfer to $l_p^n$ for $2 < p < p_0 \simeq 26.265$. Then the hyperplane section perpendicular to the main diagonal yields a counterexample for large dimensions $n$. We show that the analogue of Ball's result holds in $l_p^n$-balls for all hyperplanes with normal unit vectors $a$, if all coordinates of $a$ have modulus $\le \frac 1 {\sqrt 2}$ and $p$ has distance $\ge 2^{-p}$ to the even integers. Under similar assumptions, we give a Gaussian upper bound for $20 < p < p_0$. |
| title | On maximal hyperplane sections of the unit ball of $l_p$ for $p>2$ |
| topic | Functional Analysis 52A38, 52A40, 46B07 |
| url | https://arxiv.org/abs/2409.06432 |