Observable concentration of mm-spaces into spaces with doubling measures

Abstract

The property of measure concentration is that an arbitrary 1-Lipschitz function f : X → R on an mm-space X is almost close to a constant function. In this paper, we prove that if such a concentration phenomenon arise, then any 1-Lipschitz map f from X to a space Y with a doubling measure also concentrates to a constant map. As a corollary, we get any 1-Lipschitz map to a Riemannian manifold with a lower Ricci curvature bounds also concentrates to a constant map.

CAT(0) spaces and expanders

“…A sequence of metric measure spaces (X n , d n , μ n ) is said to L 2 -concentrate for the target space Y if for any sequence of 1-Lipschitz maps f n : X n → Y , we have X n ×X n d Y ( f n (x), f n (y)) 2 dμ n (x)dμ n (y) → 0. Funano [7,8] showed that the measure concentration (in the sense of Lévy-Milman) follows from the L 2 -concentration and that the measure concentration for the target space R is equivalent to the measure concentration when the target space is an R-tree, and is also equivalent if we replace the R-tree by a finite dimensional Hadamard manifold.…”

Concentration of maps and group actions

“…Sketch of the proof following [3] Take a maximal r 0 -separated set {ξ α } α∈A of X . We can find a decomposition {ξ α } α∈A = J 1 ∪J 2 ∪· · ·∪J k such that k ≤ C(r 0 , 5r 0 ) and each J i is 5r 0 -separated set.…”

“…Note that R-trees, doubling spaces, and metric graphs have the property (♦) (Lemma 2.7 and Theorems 3.5, 3.7, 3.17). The author proved in Funano [4] that Hadamard manifolds have the property (♦).…”

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

“…The Lévy families were first introduced and analyzed by Gromov and Milman in [10]. In previous works [2], [3], [4], [5], the author proved that if a metric space Y is either an R-tree, a doubling space, a metric graph, or a Hadamard manifold, then lim n→∞ ObsDiam Y (X n ; −κ) = 0 holds for any κ > 0 and any Lévy family {X n } ∞ n=1 . To prove these results, we needed to assume the finiteness of the dimension of the target metric spaces.…”