A family of functional inequalities: Łojasiewicz inequalities and displacement convex functions.

For displacement convex functionals in the probability space equipped with the Monge–Kantorovich metric we prove the equivalence between the gradient and functional type Łojasiewicz inequalities. We also discuss the more general case of λ -convex functions and we provide a general convergence theorem for the corresponding gradient dynamics. Specialising our results to the Boltzmann entropy, we recover Otto–Villani's theorem asserting the equivalence between logarithmic Sobolev and Talagrand's inequalities. The choice of power-type entropies shows a new equivalence between Gagliardo–Nirenberg inequality and a nonlinear Talagrand inequality. Some nonconvex results and other types of equivalences are discussed. [ABSTRACT FROM AUTHOR]