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 $$|\{y\in K: |< y,x> |\gr t\|<\cdot, x>\|_1\}|\ls\exp (-ct^2/\log^2(t+1))$$ for all $t\gr 1$, 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.