  This paper is devoted to one-dimensional interpolation
Gagliardo-Nirenberg-Sobolev inequalities. We study how various notions of
duality, transport and monotonicity of functionals along flows defined by some
nonlinear diffusion equations apply. We start by reducing the inequality to a
much simpler dual variational problem using mass transportation theory. Our
second main result is devoted to the construction of a Lyapunov functional
associated with a nonlinear diffusion equation, that provides an alternative
proof of the inequality. The key observation is that the inequality on the line
is equivalent to Sobolev's inequality on the sphere, at least when the
dimension is an integer, or to the critical interpolation inequality for the
ultraspherical operator in the general case. The time derivative of the
functional along the flow is itself very interesting. It explains the machinery
of some rigidity estimates for nonlinear elliptic equations and shows how
eigenvalues of a linearized problem enter in the computations. Notions of
gradient flows are then discussed for various notions of distances. Throughout
this paper we shall deal with two classes of inequalities corresponding either
to p>2 or to p<2. The algebraic part in the computations is very similar in
both cases, although the case p<2 is definitely less standard.
