face of a convex set, alternative definition of

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

Let $C$ be a convex subset of ${ℝ}^n$. Before we define faces,we introduce oriented hyperplanes and supporting hyperplanes.

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

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

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

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

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.

• If $F_1=C\cap H(n_1,p_1)$ and $F_2=C\cap H(n_2,p_2)$ are facesof $C$ intersecting in a point $p$, then $H(n_1+n_2,p)$ is asupporting 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.

• Since each proper face lies on the base of the upper halfspace of somesupporting hyperplane, each such face must lie on the relativeboundary of $C$.