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v2006.03.09 - Convex Optimization

v2006.03.09 - Convex Optimization

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50 CHAPTER 2. CONVEX GEOMETRY(a)R(b)R 2(c)R 3Figure 9: (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 0-dimensional faces.Intersection of line with boundary is a point at entry to interior. These samefacts hold in higher dimension.2.1.7 classical boundary(confer2.6.1.3) Boundary of a set C is the closure of C less its interiorpresumed nonempty; [36,1.1]∂ C = C \ int C (13)which follows from the factint C = C (14)assuming nonempty interior. 2.6 One implication is: an open set has aboundary defined although not contained in the set.2.6 Otherwise, for x∈ R n as in (10), [160,2.1,2.3]the empty set is both open and closed.int{x} = ∅ = ∅

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