v2010.10.26 - Convex Optimization

2.1. CONVEX SET 41(a)R(b)R 2(c)R 3Figure 13: (a) Ellipsoid in R is a line segment whose boundary comprises twopoints. Intersection of line with ellipsoid in R , (b) in R 2 , (c) in R 3 . Eachellipsoid illustrated has entire boundary constituted by zero-dimensionalfaces; in fact, by vertices (2.6.1.0.1). Intersection of line with boundaryis a point at entry to interior. These same facts hold in higher dimension.int C = C \∂ Ca bounded open set has boundary defined but not contained in the setinterior of an open set is equivalent to the set itself;from which an open set is defined: [258, p.109]C is open ⇔ int C = C (19)C is closed ⇔ int C = C (20)The set illustrated in Figure 12b is not open because it is not equivalentto its interior, for example, it is not closed because it does not containits boundary, and it is not convex because it does not contain all convexcombinations of its boundary points.the empty set is both open and closed.