  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.
