  We consider a family of Gagliardo-Nirenberg-Sobolev interpolation
inequalities which interpolate between Sobolev's inequality and the logarithmic
Sobolev inequality, with optimal constants. The difference of the two terms in
the interpolation inequalities (written with optimal constant) measures a
distance to the manifold of the optimal functions. We give an explicit estimate
of the remainder term and establish an improved inequality, with explicit norms
and fully detailed constants. Our approach is based on nonlinear evolution
equations and improved entropy - entropy production estimates along the
associated flow. Optimizing a relative entropy functional with respect to a
scaling parameter, or handling properly second moment estimates, turns out to
be the central technical issue. This is a new method in the theory of nonlinear
evolution equations, which can be interpreted as the best fit of the solution
in the asymptotic regime among all asymptotic profiles.
