v2009.01.01 - Convex Optimization

2.10. CONIC INDEPENDENCE (C.I.) 127
0
0
0
(a) (b) (c)
Figure 43: Vectors in R 2 : (a) affinely and conically independent,
(b) affinely independent but not conically independent, (c) conically
independent but not affinely independent. None of the examples exhibits
linear independence. (In general, a.i. c.i.)
2.10.0.0.1 Table: Maximum number of c.i. directions
n supk (pointed) supk (not pointed)
0 0 0
1 1 2
2 2 4
3
.
∞
.
∞
.
Assuming veracity of this table, there is an apparent vastness between two
and three dimensions. The finite numbers of conically independent directions
indicate:
Convex cones in dimensions 0, 1, and 2 must be polyhedral. (2.12.1)
Conic independence is certainly one convex idea that cannot be completely
explained by a two-dimensional picture as Barvinok suggests [25, p.vii].
From this table it is also evident that dimension of Euclidean space cannot
exceed the number of conically independent directions possible;
n ≤ supk
tangent to svec ∂ S N + at a point because all one-dimensional faces of S N + are exposed.
Because a pointed convex cone has only one vertex, the origin, there can be no intersection
of svec ∂ S N + with any higher-dimensional affine subset A that will make a nonzero point.