Convex smooth-like properties and Faces Radon–Nikodým property in Banach spaces
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. We show that many of the reductions made for the Asplund property also work for these smooth-like properties. In this framework, we introduce a new geometrical property, called the Faces Radon–Nikodým property, and we prove that it is in duality with a convex $w^*$-smooth-like property.