The Hamilton-Jacobi semigroup on length spaces and applications

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.