Push forward measures and concentration phenomena

TL;DR

This paper investigates how measure concentration properties are transferred via push-forward maps between measures, relating the concentration inequalities to Banach-Mazur distances and Lipschitz constants, with applications to normed spaces and convex bodies.

Contribution

It introduces refined concentration inequalities for push-forward measures involving Banach-Mazur distances and Lipschitz maps, with applications to normed spaces and convex bodies.

Findings

Concentration inequalities depend on medians and Banach-Mazur distances.

Normed spaces with good concentration are far from high-dimensional cubes.

Established concentration for the cross polytope with Lebesgue measure and  norm.

Abstract

In this note we study how a concentration phenomenon can be transmitted from one measure $\mu$ to a push-forward measure $\nu$. In the first part, we push forward $\mu$ by $\pi:supp(\mu)\rightarrow \Ren$, where $\pi x=\frac{x}{\norm{x}_L}\norm{x}_K$, and obtain a concentration inequality in terms of the medians of the given norms (with respect to $\mu$) 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 Banach-Mazur 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 $\mu$ and $\nu$ are given, both related to the norm $\norm{\cdot}_L$, 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 $\mu$ into $\nu$. As an application, we obtain a concentration inequality for the cross polytope with respect to the normalized Lebesgue measure and the $\ell_1$ norm.