Central and $L^p$-concentration of 1-Lipschitz maps into $\mathbb{R}$-trees

TL;DR

This paper investigates how 1-Lipschitz maps from metric measure spaces concentrate into $ ext{R}$-trees and shows that such concentration is equivalent to concentration into the real line, revealing a fundamental geometric property.

Contribution

It establishes the equivalence between concentration phenomena into $ ext{R}$-trees and the real line for 1-Lipschitz maps from mm-spaces, advancing understanding of metric measure space geometry.

Findings

Concentration to $ ext{R}$-trees is equivalent to concentration to the real line.

Main theorems demonstrate this equivalence for 1-Lipschitz maps.

Results deepen the understanding of geometric concentration phenomena.

Abstract

In this paper, we study the L\'{e}vy-Milman concentration phenomenon of 1-Lipschitz maps from mm-spaces to $\mathbb{R}$-trees. Our main theorems assert that the concentration to $\mathbb{R}$-trees is equivalent to the concentration to the real line.