Push forward measures and concentration phenomena

Abstract

Abstract. In this note we study how a concentration phenomenon can be transmitted from one measure µ to a push-forward measure ν. In the first part, we push forward µ by π :x K , and obtain a concentration inequality in terms of the medians of the given norms (with respect to µ) and the Banach-Mazur distance between them. This approach is finer than simply bounding the concentration of the push forward measure in terms of the BanachMazur distance between K and L. As a corollary we show that any normed probability space with good concentration is far from any high dimensional subspace of the cube. In the second part, two measures µ and ν are given, both related to the norm · L, obtaining a concentration inequality in which it is involved the Banach-Mazur distance between K and L and the Lipschitz constant of the map that pushes forward µ into ν. As an application, we obtain a concentration inequality for the cross polytope with respect to the normalized Lebesgue measure and the ℓ1 norm.