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In this paper, we study the L\'{e}vy-Milman concentration phenomenon of1-Lipschitz maps into infinite dimensional metric spaces. Our main theoremasserts that the concentration to an infinite dimensional $\ell^p$-ball withthe $\ell^q$-distance function for $1\leq p<q\leq +\infty$ is equivalent to theconcentration to the real line.

Concentration of 1-Lipschitz maps into an infinite dimensional $\ell^p$-ball with $\ell^q$-distance function