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

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 $1 \le p \le n$.