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

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.

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 $C$.
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