A note on subgaussian estimates for linear functionals on convex bodies

摘要

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 R-n with volume one and center of mass at the origin, there exists x not equal 0 such that vertical bar{y is an element of K : vertical bar y, x vertical bar >= t vertical bar vertical bar ., x vertical bar vertical bar 1} vertical bar <= exp(-ct(2) / log(2) (t+1)) for all t = 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.