Most Convex Functions Have Unique Minimizers

Abstract

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.