This is an announcement for the paper "Push forward measures and concentration phenomena" by C. Hugo Jimenez, Marton Naszodi, and Rafael Villa.
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.
Archive classification: math.FA
Mathematics Subject Classification: 46B06, 46b07, 46B09, 52A20
Remarks: 12 pages
Submitted from: carloshugo@us.es
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1112.4765
or