v2009.01.01 - Convex Optimization

2.8. CONE BOUNDARY 103
From Theorem 2.8.1.1.1,
rel∂C = C \ rel int C (153)
= conv{exposed points and exposed rays} \ rel int C
= conv{extreme points and extreme rays} \ rel int C
⎫
⎪⎬
⎪⎭
(172)
Thus each and every extreme point of a convex set (that is not a point)
resides on its relative boundary, while each and every extreme direction of a
convex set (that is not a halfline and contains no line) resides on its relative
boundary because extreme points and directions of such respective sets do
not belong to relative interior by definition.
The relationship between extreme sets and the relative boundary actually
goes deeper: Any face F of convex set C (that is not C itself) belongs to
rel ∂ C , so dim F < dim C . [266,18.1.3]
2.8.2.2 Converse caveat
It is inconsequent to presume that each and every extreme point and direction
is necessarily exposed, as might be erroneously inferred from the conventional
boundary definition (2.6.1.3.1); although it can correctly be inferred: each
and every extreme point and direction belongs to some exposed face.
Arbitrary points residing on the relative boundary of a convex set are not
necessarily exposed or extreme points. Similarly, the direction of an arbitrary
ray, base 0, on the boundary of a convex cone is not necessarily an exposed
or extreme direction. For the polyhedral cone illustrated in Figure 20, for
example, there are three two-dimensional exposed faces constituting the
entire boundary, each composed of an infinity of rays. Yet there are only
three exposed directions.
Neither is an extreme direction on the boundary of a pointed convex cone
necessarily an exposed direction. Lift the two-dimensional set in Figure 27,
for example, into three dimensions such that no two points in the set are
collinear with the origin. Then its conic hull can have an extreme direction
B on the boundary that is not an exposed direction, illustrated in Figure 36.