face of a convex set, alternative definition of

The following definition of a face of a convex set in a real vector space is sometimes useful.

Let C be a convex subset of n. Before we define faces, we introduce oriented hyperplanesMathworldPlanetmathPlanetmath and supporting hyperplanes.

Given any vectors n and p in n, define the hyperplane H(n,p) by


note that this is the degenerate hyperplane n if n=0. As long as H(n,p) is nondegenerate, its removal disconnects n. The upper halfspace of n determined by H(n,p) is


A hyperplane H(n,p) is a supporting hyperplane for C if its upper halfspace contains C, that is, if CH(n.p)+.

Using this terminology, we can define a face of a convex set C to be the intersection of C with a supporting hyperplane of C. Notice that we still get the empty set and C as improper faces of C.

Remarks. Let C be a convex set.

  • If F1=CH(n1,p1) and F2=CH(n2,p2) are faces of C intersecting in a point p, then H(n1+n2,p) is a supporting hyperplane of C, and F1F2=CH(n1+n2,p). This shows that the faces of C form a meet-semilattice.

  • Since each proper face lies on the base of the upper halfspace of some supporting hyperplane, each such face must lie on the relative boundary of C.

Title face of a convex set, alternative definition of
Canonical name FaceOfAConvexSetAlternativeDefinitionOf
Date of creation 2013-03-22 17:02:02
Last modified on 2013-03-22 17:02:02
Owner mps (409)
Last modified by mps (409)
Numerical id 4
Author mps (409)
Entry type Definition
Classification msc 52A99
Synonym face
Defines supporting hyperplane