Geometric Properties of Maximal Monotone Operators and Convex Functions which May Represent Them

We study the relations between some geometric properties of maximal monotone operators and generic geometric and analytical properties of the functions on the associate Fitzpatrick family of convex representations. We present some new technical properties of the Fitzpatrick families associated to bounded-range and bounded-domain maximal monotone operator which infer, among other properties, that the convex hull of the range of bounded domain maximal monotone operators are weak-* dense; we also present sufficient conditions for a convex function to represent a bounded-range maximal monotone operator; and finally give an example which makes clear that the result of Zagrodny that maximal monotone operators with a relatively compact range are of type (D) can not be extended to the weak* topology.