  In this work we consider dimensional improvements of the logarithmic Sobolev,
Talagrand and Brascamp-Lieb inequalities. For this we use optimal transport
methods and the Borell-Brascamp-Lieb inequality. These refinements can be
written as a deficit in the classical inequalities. They have the right scale
with respect to the dimension. They lead to sharpened concentration properties
as well as refined contraction bounds, convergence to equilibrium and short
time behaviour for the laws of solutions to stochastic differential equations.
