  We prove a general duality result showing that a Brascamp--Lieb type
inequality is equivalent to an inequality expressing subadditivity of the
entropy, with a complete correspondence of best constants and cases of
equality. This open a new approach to the proof of Brascamp--Lieb type
inequalities, via subadditivity of the entropy. We illustrate the utility of
this approach by proving a general inequality expressing the subadditivity
property of the entropy on $\R^n$, and fully determining the cases of equality.
As a consequence of the duality mentioned above, we obtain a simple new proof
of the classical Brascamp--Lieb inequality, and also a fully explicit
determination of all of the cases of equality. We also deduce several other
consequences of the general subadditivity inequality, including a
generalization of Hadamard's inequality for determinants. Finally, we also
prove a second duality theorem relating superadditivity of the Fisher
information and a sharp convolution type inequality for the fundamental
eigenvalues of Schr\"odinger operators. Though we focus mainly on the case of
random variables in $\R^n$ in this paper, we discuss extensions to other
settings as well.
