  We prove comparison theorems for the sub-Riemannian distortion coefficients
appearing in interpolation inequalities. These results, which are equivalent to
a sub-Laplacian comparison theorem for the sub-Riemannian distance, are
obtained by introducing a suitable notion of sub-Riemannian Bakry-\'Emery
curvature. The model spaces for comparison are variational problems coming from
optimal control theory. As an application we establish the sharp measure
contraction property for 3-Sasakian manifolds satisfying a suitable curvature
bound.
