Dimension-free Gaussian tail estimates for linear functionals on convex bodies
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| Acceso en línea: | |
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| _version_ | 1866917481157754880 |
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| author | Letwin, Brayden Mikulincer, Dan |
| author_facet | Letwin, Brayden Mikulincer, Dan |
| contents | Let $K \subset \mathbb{R}^n$ be a centered convex body of volume one. We prove that there exist absolute constants $c,C > 0$ and an orthonormal set of vectors $Θ\subset S^{n-1}$ with size $\left|Θ\right| \ge 9n/10$ such that, if $X$ is a random vector uniformly distributed on $K$, then for all $θ\in Θ$ one has \[
c\cdot \sqrt{p}\,\left(\mathbb{E} \left|\left\langle X,θ\right\rangle\right|^2\right)^{1/2}
\le
\left(\mathbb{E} \left|\left\langle X,θ\right\rangle\right|^p\right)^{1/p}
\le
C\cdot \sqrt{p}\,\left(\mathbb{E} \left|\left\langle X,θ\right\rangle\right|^2\right)^{1/2}, \] where the upper estimate holds for all $p \ge 1$ while the lower bound only holds for $1 \le p \le n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_10939 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Dimension-free Gaussian tail estimates for linear functionals on convex bodies Letwin, Brayden Mikulincer, Dan Metric Geometry Probability Let $K \subset \mathbb{R}^n$ be a centered convex body of volume one. We prove that there exist absolute constants $c,C > 0$ and an orthonormal set of vectors $Θ\subset S^{n-1}$ with size $\left|Θ\right| \ge 9n/10$ such that, if $X$ is a random vector uniformly distributed on $K$, then for all $θ\in Θ$ one has \[ c\cdot \sqrt{p}\,\left(\mathbb{E} \left|\left\langle X,θ\right\rangle\right|^2\right)^{1/2} \le \left(\mathbb{E} \left|\left\langle X,θ\right\rangle\right|^p\right)^{1/p} \le C\cdot \sqrt{p}\,\left(\mathbb{E} \left|\left\langle X,θ\right\rangle\right|^2\right)^{1/2}, \] where the upper estimate holds for all $p \ge 1$ while the lower bound only holds for $1 \le p \le n$. |
| title | Dimension-free Gaussian tail estimates for linear functionals on convex bodies |
| topic | Metric Geometry Probability |
| url | https://arxiv.org/abs/2605.10939 |