v2007.11.26 - Convex Optimization

36 CHAPTER 2. CONVEX GEOMETRY2.1.3 Orthant:name given to a closed convex set that is the higher-dimensionalgeneralization of quadrant from the classical Cartesian partition of R 2 .The most common is the nonnegative orthant R n + or R n×n+ (analogueto quadrant I) to which membership denotes nonnegative vector- ormatrix-entries respectively; e.g.,R n +∆= {x∈ R n | x i ≥ 0 ∀i} (8)The nonpositive orthant R n − or R n×n− (analogue to quadrant III) denotesnegative and 0 entries. Orthant convexity 2.3 is easily verified bydefinition (1).2.1.4 affine setA nonempty affine set (from the word affinity) is any subset of R n that is atranslation of some subspace. An affine set is convex and open so contains noboundary: e.g., ∅ , point, line, plane, hyperplane (2.4.2), subspace, etcetera.For some parallel 2.4 subspace M and any point x∈AA is affine ⇔ A = x + M= {y | y − x∈M}(9)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 the dimension of its affine hull; [282, p.14]dim S ∆ = dim aff S = dim aff(S − s) , s∈ S (10)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.[148, A.2.1]2.3 All orthants are self-dual simplicial cones. (2.13.5.1,2.12.3.1.1)2.4 Two affine sets are said to be parallel when one is a translation of the other. [232, p.4]