Moment estimates for convex measures
Résumé
Let p ≥ 1, ε > 0, r ≥ (1+ε)p, and X be a (−1/r)-concave random vector in Rn with Euclidean norm |X|. We prove that where (E|X|p)1/p ≤c(C(ε)E|X|+σp(X)), σp(X)= sup(E|⟨z,X⟩|p)1/p, |z|≤1 C(ε) depends only on ε and c is a universal constant. Moreover, if in addition X is centered then (E|X|−p)−1/p ≥c(ε)(E|X|−Cσp(X)).