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 $\mathbb{R}^{n}$ . Before we define faces, we introduce oriented hyperplanes and supporting hyperplanes.

Given any vectors $n$ and $p$ in $\mathbb{R}^{n}$ , define the hyperplane $H(n,p)$ by

$H(n,p)=\{x\in\mathbb{R}^{n}\colon n\cdot(x-p)=0\};$

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

$H(n,p)^{+}=\{x\in\mathbb{R}^{n}\colon n\cdot(x-p)\geq 0\}.$

A hyperplane $H(n,p)$ is a supporting hyperplane for $C$ if its upper halfspace contains $C$ , that is, if $C\subset H(n.p)^{+}$ .

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.

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If $F_{1}=C\cap H(n_{1},p_{1})$ and $F_{2}=C\cap H(n_{2},p_{2})$ are faces of $C$ intersecting in a point $p$ , then $H(n_{1}+n_{2},p)$ is a supporting hyperplane of $C$ , and $F_{1}\cap F_{2}=C\cap H(n_{1}+n_{2},p)$ . This shows that the faces of $C$ form a meet-semilattice.
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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