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