  Finding the minimum and the minimizers of convex functions has been of
primary concern in convex analysis since its conception. It is well-known that
if a convex function has a minimum, then that minimum is global. The
minimizers, however, may not be unique. There are certain subclasses, such as
strictly convex functions, that do have unique minimizers when the minimum
exists, but other subclasses, such as constant functions, that do not. This
paper addresses the question of how many convex functions have unique
minimizers. We show, using Baire category theory, that the set of proximal
mappings of convex functions that have a unique fixed point is generic.
Consequently, the set of classes of convex functions that have unique
minimizers is generic.
