An enhanced Baillon Haddad theorem for convex functions on convex sets.
In this paper, we prove the Baillon Haddad theorem for Gateaux differentiable convex functions defined on open convex sets of arbitrary Hilbert spaces. Formally, this result establishes that the gradient of a convex function defined on an open convex set is \beta$ Lipschitz if and only if it is $1/\beta$ cocoercive. An application to convex optimization through dynamical systems is given.