We introduce the notion of convex smooth-like (resp. $w*$-smooth-like) properties, which are a generalization of the well-known Asplund (resp. $w*$-Asplund) property. These properties allow us to have different regularities for the subdifferential of convex functions. We show that many of the reductions made for Asplund property also work for these smooth-like properties. In this framework, we introduce a new geometrical property, called the Faces Radon-Nikodym property, and we prove that it is in duality with a convex $w*$-smooth-like property.