v2007.09.13 - Convex Optimization

68 CHAPTER 2. CONVEX GEOMETRYCH −H +0 > κ 3 > κ 2 > κ 1{z ∈ R 2 | a T z = κ 1 }a{z ∈ R 2 | a T z = κ 2 }{z ∈ R 2 | a T z = κ 3 }Figure 19: Each shaded line segment {z ∈ C | a T z = κ i } belonging to setC ⊂ R 2 shows intersection with hyperplane parametrized by scalar κ i ; eachshows a (linear) contour in vector z of equal inner product with normalvector a . Cartesian axes drawn for reference. (confer Figure 55)2.4.2.5 affine mapsAffine transformations preserve affine hulls. Given any affine mapping T ofvector spaces and some arbitrary set C [228, p.8]aff(T C) = T(aff C) (107)2.4.2.6 PRINCIPLE 2: Supporting hyperplaneThe second most fundamental principle of convex geometry also follows fromthe geometric Hahn-Banach theorem [181,5.12] [16,1] that guaranteesexistence of at least one hyperplane in R n supporting a convex set (having