The Sinkhorn algorithm is the nowadays most used method to evaluate the value of the optimal transport problem in the general case. From the numerics it is known that empirically the convergence is linear with a rate of convergence which is algebraic in $\ep$; in this talk we will improve on the result by Carlier on the optimal rate of convergence, under the assumption that the marginal measures satisfy a Poincaré inequality. We will discuss also counterexamples, leaving some open questions on the two-scale behavior of this optimization algorithm.