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

TL;DR

This paper establishes dimension-free tail estimates for linear functionals on convex bodies, showing that most directions exhibit Gaussian-like behavior with universal constants.

Contribution

It proves that for most directions, the moments of linear functionals on convex bodies are comparable to Gaussian moments with universal constants, independent of dimension.

Findings

Most directions form an orthonormal set covering 90% of the sphere.

Moments of linear functionals follow Gaussian tail estimates with universal constants.

Upper bounds hold for all p ≥ 1, lower bounds for 1 ≤ p ≤ n.

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 $\Theta \subset S^{n-1}$ with size $\left|\Theta\right| \ge 9n/10$ such that, if $X$ is a random vector uniformly distributed on $K$, then for all $\theta \in \Theta$ one has \[ c\cdot \sqrt{p}\,\left(\mathbb{E} \left|\left\langle X,\theta \right\rangle\right|^2\right)^{1/2} \le \left(\mathbb{E} \left|\left\langle X,\theta \right\rangle\right|^p\right)^{1/p} \le C\cdot \sqrt{p}\,\left(\mathbb{E} \left|\left\langle X,\theta \right\rangle\right|^2\right)^{1/2}, \] where the upper estimate holds for all $p \ge 1$ while the lower bound only holds for $1 \le p \le n$.