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
$$H(n,p)=\{x\in {\mathbb{R}}^{n}:n\cdot (xp)=0\};$$ 
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
$$H{(n,p)}^{+}=\{x\in {\mathbb{R}}^{n}:n\cdot (xp)\ge 0\}.$$ 
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 meetsemilattice.

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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  20130322 17:02:02 
Last modified on  20130322 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 