v2007.09.13 - Convex Optimization

2.10. CONIC INDEPENDENCE (C.I.) 121000(a) (b) (c)Figure 37: Vectors in R 2 : (a) affinely and conically independent,(b) affinely independent but not conically independent, (c) conicallyindependent but not affinely independent. None of the examples exhibitslinear independence. (In general, a.i. c.i.)2.10.0.0.1 Table: Maximum number of c.i. directionsn supk (pointed) supk (not pointed)0 0 01 1 22 2 43.∞.∞.Assuming veracity of this table, there is an apparent vastness between twoand three dimensions. The finite numbers of conically independent directionsindicate:Convex cones in dimensions 0, 1, and 2 must be polyhedral. (2.12.1)Conic independence is certainly one convex idea that cannot be completelyexplained by a two-dimensional picture. [20, p.vii]From this table it is also evident that dimension of Euclidean space cannotexceed the number of conically independent directions possible;n ≤ supktangent 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 intersectionof svec ∂ S N + with any higher-dimensional affine subset A that will make a nonzero point.