A note on subgaussian estimates for linear functionals on convex bodies

We give an alternative proof of a recent result of Klartag on the existence of almost subgaussian linear functionals on convex bodies. If $K$ is a convex body in ${\mathbb{R}}^n$ with volume one and center of mass at the origin, there exists $x\neq 0$ such that

$\vert\{ y\in K:\,\vert\langle y,x\rangle \vert\geq t\Vert\langle\cdot ,x\rangle\Vert _1\}\vert\leq\exp (-ct^2/\log^2(t+1))$

for all $t\geq 1$, where $c>0$ is an absolute constant. The proof is based on the study of the $L_q$-centroid bodies of $K$. Analogous results hold true for general log-concave measures.