Convex function, their extensions and extremal structure of their epigraphs
Date
2013-02-04
Authors
Nthebe, Johannes. M. T.
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Abstract
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.