  The goal of this note is to state the optimal decay rate for solutions of the
nonlinear fast diffusion equation and, in self-similar variables, the optimal
convergence rates to Barenblatt self-similar profiles and their
generalizations. It relies on the identification of the optimal constants in
some related Hardy-Poincar\'e inequalities and concludes a long series of
papers devoted to generalized entropies, functional inequalities and rates for
nonlinear diffusion equations.
