  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.
