v2010.10.26 - Convex Optimization

38 CHAPTER 2. CONVEX GEOMETRYFigure 11: A slab is a convex Euclidean body infinite in extent but notaffine. Illustrated in R 2 , it may be constructed by intersecting two opposinghalfspaces whose bounding hyperplanes are parallel but not coincident.Because number of halfspaces used in its construction is finite, slab is apolyhedron (2.12). (Cartesian axes + and vector inward-normal, to eachhalfspace-boundary, are drawn for reference.)2.1.4 affine setA nonempty affine set (from the word affinity) is any subset of R n that is atranslation of some subspace. Any affine set is convex and open so containsno boundary: e.g., empty set ∅ , point, line, plane, hyperplane (2.4.2),subspace, etcetera. For some parallel 2.5 subspace R and any point x ∈ AA is affine ⇔ A = x + R= {y | y − x∈R}(10)The intersection of an arbitrary collection of affine sets remains affine. Theaffine hull of a set C ⊆ R n (2.3.1) is the smallest affine set containing it.2.1.5 dimensionDimension of an arbitrary set S is Euclidean dimension of its affine hull;[371, p.14]dim S dim aff S = dim aff(S − s) , s∈ S (11)the same as dimension of the subspace parallel to that affine set aff S whennonempty. Hence dimension (of a set) is synonymous with affine dimension.[199, A.2.1]2.5 Two affine sets are said to be parallel when one is a translation of the other. [307, p.4]