Convex function, their extensions and extremal structure of their epigraphs
Nthebe, Johannes. M. T.
Let f be a real valued function with the domain dom(f) in some vector space X and let C be the collection of convex subsets of X. The following two questions are investigated; 1. Do there exist maximal convex restrictions g of f with dom(g) 2 C? 2. If f is convex with dom(f) 2 C, do there exist maximal convex extension g of f with dom(g) 2 C? We will show that the answer to both questions is positive under a certain condition on C. We also show that the extreme points of the epigraph of a real continuous strictly convex function are dense in the graph of such a function, and the set of such extreme points of an epigraph may be equal to the graph. Moreover we show that a set of extreme points of an epigraph may be equal to a graph of such a convex function under certain conditions. We also discuss conditions under which an epigraph of a real convex function on a Banach space X may, and may not, have extreme points, denting points and/or strongly exposed points. One of the interesting results in this discussion is that boundary points, extreme points, denting points and the graphs in an closed epigraph of a strictly convex function coincide. Moreover, we show that there is relationship between the extremal structure of an epigraph of a convex function and a point in a domain on which such a function attains its minimum.