v2009.01.01 - Convex Optimization

2.8. CONE BOUNDARY 101
2.8.2 Exposed direction
2.8.2.0.1 Definition. Exposed point & direction of pointed convex cone.
[266,18] (confer2.6.1.0.1)
When a convex cone has a vertex, an exposed point, it resides at the
origin; there can be only one.
In the closure of a pointed convex cone, an exposed direction is the
direction of a one-dimensional exposed face that is a ray emanating
from the origin.
{exposed directions} ⊆ {extreme directions}
△
For a proper cone in vector space R n with n ≥ 2, we can say more:
{exposed directions} = {extreme directions} (171)
It follows from Lemma 2.8.0.0.1 for any pointed closed convex cone, there
is one-to-one correspondence of one-dimensional exposed faces with exposed
directions; id est, there is no one-dimensional exposed face that is not a ray
base 0.
The pointed closed convex cone EDM 2 , for example, is a ray in
isomorphic subspace R whose relative boundary (2.6.1.3.1) is the origin.
The conventionally exposed directions of EDM 2 constitute the empty set
∅ ⊂ {extreme direction}. This cone has one extreme direction belonging to
its relative interior; an idiosyncrasy of dimension 1.
2.8.2.1 Connection between boundary and extremes
2.8.2.1.1 Theorem. Exposed. [266,18.7] (confer2.8.1.1.1)
Any closed convex set C containing no lines (and whose dimension is at
least 2) can be expressed as the closure of the convex hull of its exposed
points and exposed rays.
⋄