Observable concentration of mm-spaces into spaces with doubling measures

Details Observable concentration of mm-spaces into spaces with doubling measures Observable concentration of mm-spaces into spaces with doubling measures

2007-01-19

arxiv.org/abs/math/0701534v1

Mathematics

Metric Geometry

8pages

Scientific paper

The property of measure concentration is that an arbitrary 1-Lipschitz function $f:X\to \mathbb{R}$ 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.

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