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

Abstract

In this paper, we study the Lévy-Milman concentration phenomenon of 1-Lipschitz maps into infinite dimensional metric spaces. Our main theorem asserts that the concentration to an infinite dimensional ℓ p -ball with the ℓ q -distance function for 1 ≤ p < q ≤ +∞ is equivalent to the concentration to the real line.