face of a convex set

Let C be a convex set in n (or any topological vector spaceMathworldPlanetmath). A face of C is a subset F of C such that

  1. 1.

    F is convex, and

  2. 2.

    given any line segmentMathworldPlanetmath LC, if ri(L)F, then LF.

Here, ri(L) denotes the relative interior of L (open segment of L).

A zero-dimensional face of a convex set C is called an extreme pointPlanetmathPlanetmath of C.

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 2 is its face (and an extreme point).

Observe that the empty setMathworldPlanetmath and C itself are faces of C. These faces are sometimes called improper faces, while other faces are called proper faces.

Remarks. Let C be a convex set.

  • The intersectionMathworldPlanetmath of two faces of C is a face of C.

  • A face of a face of C is a face of C.

  • Any proper face of C lies on its relative boundary, rbd(C).

  • The set Part(C) of all relative interiors of the faces of C partitions C.

  • If C is compactPlanetmathPlanetmath, then C is the convex hullMathworldPlanetmath of its extreme points.

  • The set F(C) of faces of a convex set C forms a latticeMathworldPlanetmath, where the meet is the intersection: F1F2:=F1F2; the join of F1,F2 is the smallest face FF(C) containing both F1 and F2. This lattice is bounded latticeMathworldPlanetmath (by and C). And it is not hard to see that F(C) is a complete latticeMathworldPlanetmath.

  • However, in general, F(C) is not a modular latticeMathworldPlanetmath. As a counterexample, consider the unit square [0,1]×[0,1] and faces a=(0,0), b={(0,y)y[0,1]}, and c=(1,1). We have ab. However, a(bc)=(0,0)=(0,0), whereas (ab)c=b=.

  • Nevertheless, F(C) is a complemented latticeMathworldPlanetmath. Pick any face FF(C). If F=C, then is a complementMathworldPlanetmath of F. Otherwise, form Part(C) and Part(F), the partitions of C and F into disjoint unionsMathworldPlanetmath of the relative interiors of their corresponding faces. Clearly Part(F)Part(C) strictly. Now, it is possible to find an extreme point p such that {p}Part(C)-Part(F). Otherwise, all extreme points lie in Part(F), which leads to

    Part(F)=Part(convex hull of extreme points of C)=Part(C),

    a contradictionMathworldPlanetmathPlanetmath. Finally, let G be the convex hull of extreme points of C not contained in Part(F). We assert that G is a complement of F. If xGF, then GF is a proper face of G and of F, hence its extreme points are also extreme points of G, and of F, which is impossible by the construction of G. Therefore FG=. Next, note that the union of extreme points of G and of F is the collectionMathworldPlanetmath of all extreme points of C, this is again the result of the construction of G, so any yC is in the join of all its extreme points, which is equal to the join of F and G (since join is universally associative).

  • Additionally, in F(C), zero-dimensional faces are compact elements, and compact elements are faces with finitely many extreme points. The unit disk D is not compact in F(D). Since every face is the convex hull (join) of all extreme points it contains, F(C) is an algebraic lattice.


Title face of a convex set
Canonical name FaceOfAConvexSet
Date of creation 2013-03-22 16:23:08
Last modified on 2013-03-22 16:23:08
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 12
Author CWoo (3771)
Entry type Definition
Classification msc 52A99
Related topic ExtremePoint
Defines face
Defines proper face
Defines extreme point
Defines improper face