Mathematics – Functional Analysis
Scientific paper
2011-12-20
Mathematics
Functional Analysis
12 pages
Scientific paper
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.
Jimenez Hugo C.
NaszÓdi MÁrton
Villa Rafael
No associations
LandOfFree
Push forward measures and concentration phenomena does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Push forward measures and concentration phenomena, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Push forward measures and concentration phenomena will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-58675