Dimension-free Gaussian tail estimates for linear functionals on convex bodies

Abstract

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 $\hat{I}˜\subset S^{n-1}$ with size $\left|\hat{I}˜\right|\ge 9n/10$ such that, if $X$ is a random vector uniformly distributed on $K$, then for all $\hat{I}¸\in \hat{I}˜$ 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 $1\le p\le n$.

18 pages