v2009.01.01 - Convex Optimization

90 CHAPTER 2. CONVEX GEOMETRY
X
X
Figure 32: Truncated nonconvex cone X = {x ∈ R 2 | x 1 ≥ x 2 , x 1 x 2 ≥ 0}.
Boundary is also a cone. [215,2.4] Cartesian axes drawn for reference. Each
half (about the origin) is itself a convex cone.
The conventional boundary of a single ray, base 0, in any dimension is
the origin because that is the union of all exposed faces not containing the
entire set. Its relative interior is the ray itself excluding the origin.
2.7.1 Cone defined
A set X is called, simply, cone if and only if
Γ ∈ X ⇒ ζΓ ∈ X for all ζ ≥ 0 (156)
where X denotes closure of cone X . An example of such a cone is the union
of two opposing quadrants; e.g., X ={x∈ R 2 | x 1 x 2 ≥0} which is not convex.
[324,2.5] Similar examples are shown in Figure 28 and Figure 32.
All cones can be defined by an aggregate of rays emanating exclusively
from the origin (but not all cones are convex). Hence all closed cones contain
the origin and are unbounded, excepting the simplest cone {0}. The empty
set ∅ is not a cone, but its conic hull is;
cone ∅ ∆ = {0} (95)