Thin-shell bounds via parallel coupling
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866915811609804800 |
|---|---|
| author | Klartag, Boaz Lehec, Joseph |
| author_facet | Klartag, Boaz Lehec, Joseph |
| contents | We prove that for any log-concave random vector $X$ in $\mathbb{R}^n$ with mean zero and identity covariance, $$ \mathbb{E} (|X| - \sqrt{n})^2 \leq C $$ where $C > 0$ is a universal constant. Thus, most of the mass of the random vector $X$ is concentrated in a thin spherical shell, whose width is only $C / \sqrt{n}$ times its radius. This confirms the thin-shell conjecture in high dimensional convex geometry. Our method relies on the construction of a certain coupling between log-affine perturbations of the law of $X$ related to Eldan's stochastic localization and to the theory of non-linear filtering. Another ingredient is a recent breakthrough technique by Guan that was previously used in our proof of Bourgain's slicing conjecture, which is known to be implied by the thin-shell conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_15495 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Thin-shell bounds via parallel coupling Klartag, Boaz Lehec, Joseph Probability Functional Analysis Metric Geometry 52A23, 39B62, 60H30 We prove that for any log-concave random vector $X$ in $\mathbb{R}^n$ with mean zero and identity covariance, $$ \mathbb{E} (|X| - \sqrt{n})^2 \leq C $$ where $C > 0$ is a universal constant. Thus, most of the mass of the random vector $X$ is concentrated in a thin spherical shell, whose width is only $C / \sqrt{n}$ times its radius. This confirms the thin-shell conjecture in high dimensional convex geometry. Our method relies on the construction of a certain coupling between log-affine perturbations of the law of $X$ related to Eldan's stochastic localization and to the theory of non-linear filtering. Another ingredient is a recent breakthrough technique by Guan that was previously used in our proof of Bourgain's slicing conjecture, which is known to be implied by the thin-shell conjecture. |
| title | Thin-shell bounds via parallel coupling |
| topic | Probability Functional Analysis Metric Geometry 52A23, 39B62, 60H30 |
| url | https://arxiv.org/abs/2507.15495 |