An enhanced Baillon-Haddad theorem for convex functions on convex sets

The Baillon-Haddad theorem establishes that the gradient of a convex and continuously differentiable function defined in a Hilbert space is β-Lipschitz if and only if it is 1/β-cocoercive. In this paper, we extend this theorem to Gateaux differentiable convex functions defined on an open convex set of a Hilbert space. Finally, we give a characterization of C1,+ convex functions in terms of local cocoercitivity.