chp1 - Lecture Notes in Convexity Instructor: Renato D. C....

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Unformatted text preview: Lecture Notes in Convexity Instructor: Renato D. C. Monteiro November 27, 2010 Chapter 1 Convex sets 1.1 Notation In this section, we introduce some global notation and terminology that will be used throughout our presentation. We write V W to indicate that the set V is a subset of the set W . We write x V (resp., x negationslash V ) to indicate that x is (resp., is not) an element of the set V . We denote the sets of real numbers, real n-vectors and real m n matrices by , n and m n , respectively. For scalars , define ( , ) := { t : < t < } , [ , ] := { t : t } , [ , ) := { t : t < } and ( , ] := { t : < t } . Given two distinct vectors x,y n , define [ x,y ] := { (1 ) x + y : [0 , 1] } , ( x,y ] := { (1 ) x + y : (0 , 1] } , [ x,y ) := { (1 ) x + y : [0 , 1) } , ( x,y ) := { (1 ) x + y : (0 , 1) } . Note that if x = y , then all the above sets reduce to the singleton { x } = { y } . Otherwise, they describe the four possible segments connecting x and y , which depend on whether the endpoints x and y are included or not in the segment. 1.2 Subspaces and Affine Manifolds def:subspace Definition 1.2.1 A nonempty subset V n is a subspace if it is closed under addition and scalar multiplication, that is, x + 1 x 1 V for every x ,x 1 V and , 1 . exerc:sub Exercise 1.2.2 Show that a nonempty subset V n is a subspace if, and only if, for every x,y V and , the vectors x + y , x and x are in V . :affine-manifold Definition 1.2.3 A subset V n is an affine manifold if x +(1 ) x 1 V for every x ,x 1 V and . In other words, a subset V of n is an affine manifold if for any two elements x ,x 1 of V , the line passing through x and x 1 is in V . Singletons, lines, and n itself are examples of affine manifolds. 1 x V V - {x } Figure 1.1: The set V is an affine manifold, V { x } is a subspace, for some x V . fig:subspace It is easy to see that a subset of n is a subspace if, and only if, it is an affine manifold containing the origin. Exercise 1.2.4 Let V n be an affine manifold. Prove that for every x V , the set V { x } is a subspace which does not depend on x (see Figure 1.1). Definition 1.2.5 The dimension of an affine manifold V n is the dimension of the subspace V x , where x is an arbitrary point in V . Definition 1.2.6 A : n m is an affine map if, for every x ,x 1 n and : A ( x + (1 ) x 1 ) = A ( x ) + (1 ) A ( x 1 ) ....
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chp1 - Lecture Notes in Convexity Instructor: Renato D. C....

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