v2010.10.26 - Convex Optimization

142 CHAPTER 2. CONVEX GEOMETRY000(a) (b) (c)Figure 49: 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.)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 as Barvinok suggests [26, p.vii].From this table it is also evident that dimension of Euclidean space cannotexceed the number of conically independent directions possible;n ≤ supk2.10.1 Preservation of conic independenceIndependence in the linear (2.1.2.1), affine (2.4.2.4), and conic senses canbe preserved under linear transformation. Suppose a matrix X ∈ R n×N (280)holds a conically independent set columnar. Consider a transformation onthe domain of such matricesT(X) : R n×N → R n×N XY (283)where fixed matrix Y [y 1 y 2 · · · y N ]∈ R N×N represents linear operator T .Conic independence of {Xy i ∈ R n , i=1... N} demands, by definition (279),Xy i ζ i + · · · + Xy j ζ j − Xy l = 0, i≠ · · · ≠j ≠l = 1... N (284)