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 Rn with volume one and center of mass at the origin, there exists x 6= 0 such that |{y ∈ K : |〈y, x〉| > t‖〈·, x〉‖1}| 6 exp(−ct 2/ log2(t+ 1)) for all t > 1, where c > 0 is an absolute constant. The proof is based on the study of the Lq–centroid bodies of K. Analogous results hold true for general log-concave measures.