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[parent] alternative definition of face of a convex set (Definition)

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

$\displaystyle 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
$\displaystyle H(n,p)^+ = \{ x \in \mathbb{R}^n \colon 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.



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Other names:  face
Also defines:  supporting hyperplane

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Cross-references: relative boundary, lie on, lies on, proper face, meet-semilattice, point, improper faces, empty set, intersection, convex set, contains, nondegenerate, vectors, hyperplanes, oriented, convex subset, vector space, real, face of a convex set
There are 39 references to this entry.

This is version 1 of alternative definition of face of a convex set, born on 2007-05-02.
Object id is 9320, canonical name is AlternativeDefinitionOfFaceOfAConvexSet.
Accessed 1291 times total.

Classification:
AMS MSC: 52A99 (Convex and discrete geometry :: General convexity :: Miscellaneous)
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