  In this paper, we study the L\'{e}vy-Milman concentration phenomenon of
1-Lipschitz maps into infinite dimensional metric spaces. Our main theorem
asserts that the concentration to an infinite dimensional $\ell^p$-ball with
the $\ell^q$-distance function for $1\leq p<q\leq +\infty$ is equivalent to the
concentration to the real line.
