ψα-ESTIMATES FOR MARGINALS OF LOG-CONCAVE PROBABILITY MEASURES

We show that a random marginal pi F(mu) of an isotropic log-concave probability measure mu on R-n exhibits better psi(alpha)-behavior. For a natural variant psi'(alpha) of the standard psi(alpha)-norm we show the following: (i) If k <= root n, then for a random F epsilon G(n,k) we have that pi F(mu) is a psi'(2) measure. We complement this result by showing that a random pi F(mu) is, at the same time, super-Gaussian. (ii) If k = n(delta), 1/2 < delta < 1, then for a random F epsilon G(n,k) we have that pi F(mu) is a psi'(alpha)(delta) = 2 delta/3 delta-1