v2010.10.26 - Convex Optimization

2.4. HALFSPACE, HYPERPLANE 792.4.2.4 Preservation of affine independenceIndependence in the linear (2.1.2.1), affine, and conic (2.10.1) senses canbe preserved under linear transformation. Suppose a matrix X ∈ R n×N (76)holds an affinely independent set in its columns. Consider a transformationon the domain of such matricesT(X) : R n×N → R n×N XY (125)where fixed matrix Y [y 1 y 2 · · · y N ]∈ R N×N represents linear operator T .Affine independence of {Xy i ∈ R n , i=1... N} demands (by definition (123))there exist no solution ζ ∈ R N ζ T 1=1, ζ k = 0, toXy i ζ i + · · · + Xy j ζ j − Xy k = 0, i≠ · · · ≠j ≠k = 1... N (126)By factoring out X , we see that is ensured by affine independence of{y i ∈ R N } and by R(Y )∩ N(X) = 0 where2.4.2.5 Affine mapsN(A) = {x | Ax=0} (143)Affine transformations preserve affine hulls. Given any affine mapping T ofvector spaces and some arbitrary set C [307, p.8]aff(T C) = T(aff C) (127)2.4.2.6 PRINCIPLE 2: Supporting hyperplaneThe second most fundamental principle of convex geometry also follows fromthe geometric Hahn-Banach theorem [250,5.12] [18,1] that guaranteesexistence of at least one hyperplane in R n supporting a full-dimensionalconvex set 2.18 at each point on its boundary.The partial boundary ∂H of a halfspace that contains arbitrary set Y iscalled a supporting hyperplane ∂H to Y when the hyperplane contains atleast one point of Y . [307,11]2.18 It is customary to speak of a hyperplane supporting set C but not containing C ;called nontrivial support. [307, p.100] Hyperplanes in support of lower-dimensional bodiesare admitted.