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

TL;DR

This paper investigates how 1-Lipschitz maps into infinite dimensional $ ext{ell}^p$-balls with $ ext{ell}^q$-distances exhibit concentration phenomena, showing equivalence to concentration on the real line.

Contribution

It establishes a new equivalence between concentration in infinite dimensional $ ext{ell}^p$-balls and the real line for certain $p$ and $q$ values.

Findings

Concentration to infinite dimensional $ ext{ell}^p$-balls is equivalent to real line concentration.

The result applies for $1 \\leq p < q \\leq +\\infty$.

Provides insight into the geometry of high-dimensional metric spaces.

Abstract

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.