  The Poincare constant R(Y) of a random variable Y relates the L2 norm of a
function g and its derivative g'. Since R(Y) - Var(Y) is positive, with
equality if and only if Y is normal, it can be seen as a distance from the
normal distribution. In this paper we establish a best possible rate of
convergence of this distance in the Central Limit Theorem. Furthermore, we show
that R(Y) is finite for discrete mixtures of normals, allowing us to add rates
to the proof of the Central Limit Theorem in the sense of relative entropy.
