  The Fisher infinitesimal model is a classical model of phenotypic trait
inheritance in quantitative genetics. Here, we prove that it encompasses a
remarkable convexity structure which is compatible with a selection function
having a convex shape. It yields uniform contractivity along the flow, as
measured by a $L^\infty$ version of the Fisher information. It induces in turn
asynchronous exponential growth of solutions, associated with a well-defined,
log-concave, equilibrium distribution. Although the equation is non-linear and
non-conservative, our result shares some similarities with the Bakry-Emery
approach to the exponential convergence of solutions to the Fokker-Planck
equation with a convex potential. Indeed, the contraction takes place at the
level of the Fisher information. Moreover, the key lemma for proving
contraction involves the Wasserstein distance $W_\infty$ between two
probability distributions of a (dual) backward-in-time process, and it is
inspired by a maximum principle by Caffarelli for the Monge-Amp\`ere equation.
