Resilience of cube slicing in $\ell_p$
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2022
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866909465781993472 |
|---|---|
| author | Eskenazis, Alexandros Nayar, Piotr Tkocz, Tomasz |
| author_facet | Eskenazis, Alexandros Nayar, Piotr Tkocz, Tomasz |
| contents | Ball's celebrated cube slicing (1986) asserts that among hyperplane sections of the cube in $\mathbb{R}^n$, the central section orthogonal to $(1,1,0,\dots,0)$ has the greatest volume. We show that the same continues to hold for slicing $\ell_p$ balls when $p > 10^{15}$, as well as that the same hyperplane minimizes the volume of projections of $\ell_q$ balls for $1 < q < 1 + 10^{-12}$. This extends Szarek's optimal Khinchin inequality (1976) which corresponds to $q=1$. These results thus address the resilience of the Ball--Szarek hyperplane in the ranges $2 < p < \infty$ and $1 < q < 2$, where analysis of the extremizers has been elusive since the works of Koldobsky (1998), Barthe--Naor (2002) and Oleszkiewicz (2003). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_01986 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Resilience of cube slicing in $\ell_p$ Eskenazis, Alexandros Nayar, Piotr Tkocz, Tomasz Functional Analysis Metric Geometry Probability 52A40, 52A20, 52A38, 60E15 Ball's celebrated cube slicing (1986) asserts that among hyperplane sections of the cube in $\mathbb{R}^n$, the central section orthogonal to $(1,1,0,\dots,0)$ has the greatest volume. We show that the same continues to hold for slicing $\ell_p$ balls when $p > 10^{15}$, as well as that the same hyperplane minimizes the volume of projections of $\ell_q$ balls for $1 < q < 1 + 10^{-12}$. This extends Szarek's optimal Khinchin inequality (1976) which corresponds to $q=1$. These results thus address the resilience of the Ball--Szarek hyperplane in the ranges $2 < p < \infty$ and $1 < q < 2$, where analysis of the extremizers has been elusive since the works of Koldobsky (1998), Barthe--Naor (2002) and Oleszkiewicz (2003). |
| title | Resilience of cube slicing in $\ell_p$ |
| topic | Functional Analysis Metric Geometry Probability 52A40, 52A20, 52A38, 60E15 |
| url | https://arxiv.org/abs/2211.01986 |