We analyze the rate of convergence towards self-similarity for the
subcritical Keller-Segel system in the radially symmetric two-dimensional case
and in the corresponding one-dimensional case for logarithmic interaction. We
measure convergence in Wasserstein distance. The rate of convergence towards
self-similarity does not degenerate as we approach the critical case. As a
byproduct, we obtain a proof of the logarithmic Hardy-Littlewood-Sobolev
inequality in the one dimensional and radially symmetric two dimensional case
based on optimal transport arguments. In addition we prove that the
one-dimensional equation is a contraction with respect to Fourier distance in
the subcritical case
