v2007.09.13 - Convex Optimization

34 CHAPTER 2. CONVEX GEOMETRY2.1 Convex setA set C is convex iff for all Y, Z ∈ C and 0≤µ≤1µ Y + (1 − µ)Z ∈ C (1)Under that defining condition on µ , the linear sum in (1) is called a convexcombination of Y and Z . If Y and Z are points in Euclidean real vectorspace R n or R m×n (matrices), then (1) represents the closed line segmentjoining them. All line segments are thereby convex sets. Apparent from thedefinition, a convex set is a connected set. [188,3.4,3.5] [30, p.2]An ellipsoid centered at x = a (Figure 10, p.38), given matrix C ∈ R m×n{x∈ R n | ‖C(x − a)‖ 2 = (x − a) T C T C(x − a) ≤ 1} (2)is a good icon for a convex set.2.1.1 subspaceA nonempty subset of Euclidean real vector space R n is called a subspace(formally defined in2.5) if every vector 2.1 of the form αx + βy , forα,β∈R , is in the subset whenever vectors x and y are. [181,2.3] Asubspace is a convex set containing the origin 0, by definition. [228, p.4]Any subspace is therefore open in the sense that it contains no boundary,but closed in the sense [188,2]It is not difficult to showR n + R n = R n (3)R n = −R n (4)as is true for any subspace R , because x∈ R n ⇔ −x∈ R n .The intersection of an arbitrary collection of subspaces remains asubspace. Any subspace not constituting the entire ambient vector space R nis a proper subspace; e.g., 2.2 any line through the origin in two-dimensionalEuclidean space R 2 . The vector space R n is itself a conventional subspace,inclusively, [165,2.1] although not proper.2.1 A vector is assumed, throughout, to be a column vector.2.2 We substitute the abbreviation e.g. in place of the Latin exempli gratia.