v2010.10.26 - Convex Optimization

36 CHAPTER 2. CONVEX GEOMETRY2.1 Convex setA set C is convex iff for all Y, Z ∈ C and 0≤µ≤1µY + (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 real finite-dimensionalEuclidean vector space R n or R m×n (matrices), then (1) represents theclosed line segment joining them. All line segments are thereby convex sets.Apparent from the definition, a convex set is a connected set. [258,3.4,3.5][42, p.2] A convex set can, but does not necessarily, contain the origin 0.An ellipsoid centered at x = a (Figure 13, p.41), 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. 2.12.1.1 subspaceA nonempty subset R of real Euclidean vector space R n is called a subspace(2.5) if every vector 2.2 of the form αx + βy , for α,β∈R , is in R whenevervectors x and y are. [250,2.3] A subspace is a convex set containing theorigin, by definition. [307, p.4] Any subspace is therefore open in the sensethat it contains no boundary, but closed in the sense [258,2]It is not difficult to showR + R = R (3)R = −R (4)as is true for any subspace R , because x∈ R ⇔ −x∈ R . Given any x∈ RR = x + R (5)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., 2.3 any line through the origin in two-dimensionalEuclidean space R 2 . The vector space R n is itself a conventional subspace,inclusively, [227,2.1] although not proper.2.1 This particular definition is slablike (Figure 11) in R n when C has nontrivial nullspace.2.2 A vector is assumed, throughout, to be a column vector.2.3 We substitute abbreviation e.g. in place of the Latin exempli gratia.