face of a convex set
is convex, and
given any line segment , if , then .
This definition formalizes the notion of a face of a convex polygon or a convex polytope and generalizes it to an arbitrary convex set. For example, any point on the boundary of a closed unit disk in is its face (and an extreme point).
Remarks. Let be a convex set.
The intersection of two faces of is a face of .
A face of a face of is a face of .
Any proper face of lies on its relative boundary, .
The set of all relative interiors of the faces of partitions .
Nevertheless, is a complemented lattice. Pick any face . If , then is a complement of . Otherwise, form and , the partitions of and into disjoint unions of the relative interiors of their corresponding faces. Clearly strictly. Now, it is possible to find an extreme point such that . Otherwise, all extreme points lie in , which leads to
a contradiction. Finally, let be the convex hull of extreme points of not contained in . We assert that is a complement of . If , then is a proper face of and of , hence its extreme points are also extreme points of , and of , which is impossible by the construction of . Therefore . Next, note that the union of extreme points of and of is the collection of all extreme points of , this is again the result of the construction of , so any is in the join of all its extreme points, which is equal to the join of and (since join is universally associative).
- 1 R.T. Rockafellar, Convex Analysis, Princeton University Press, 1996.
|Title||face of a convex set|
|Date of creation||2013-03-22 16:23:08|
|Last modified on||2013-03-22 16:23:08|
|Last modified by||CWoo (3771)|