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 $\beta$-Lipschitz if and only if it is $1/\beta$-cocoercive. In this paper, we extend this theorem to G\^{a}teaux differentiable convex functions defined on an open convex set of a Hilbert space. Finally, we give a characterization of $C^{1,+}$ convex functions in terms of local cocoercitivity.