%0 Journal Article
%T The Hamilton-Jacobi semigroup on length spaces and applications
%A John Lott
%A Cedric Villani
%J Mathematics
%D 2006
%I arXiv
%X We define a Hamilton-Jacobi semigroup acting on continuous functions on a compact length space. Following a strategy of Bobkov, Gentil and Ledoux, we use some basic properties of the semigroup to study geometric inequalities related to concentration of measure. Our main results are that (1) a Talagrand inequality on a measured length space implies a global Poincare inequality and (2) if the space satisfies a doubling condition, a local Poincare inequality and a log Sobolev inequality then it also satisfies a Talagrand inequality.
%U http://arxiv.org/abs/math/0612560v2