  We prove uniqueness in the class of integrable and bounded nonnegative
solutions in the energy sense to the Keller-Segel (KS) chemotaxis system. Our
proof works for the fully parabolic KS model, it includes the classical
parabolic-elliptic KS equation as a particular case, and it can be generalized
to nonlinear diffusions in the particle density equation as long as the
diffusion satisfies the classical McCann displacement convexity condition. The
strategy uses Quasi-Lipschitz estimates for the chemoattractant equation and
the above-the-tangent characterizations of displacement convexity. As a
consequence, the displacement convexity of the free energy functional
associated to the KS system is obtained from its evolution for bounded
integrable initial data.
