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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 nvectors and real m × n matrices by ℜ , ℜ n and ℜ m × n , respectively. For scalars α ≤ β , define ( α,β ) := { t ∈ ℜ : α < t < β } , [ α,β ] := { t ∈ ℜ : α ≤ t ≤ β } , [ α,β ) := { t ∈ ℜ : α ≤ t < β } and ( α,β ] := { t ∈ ℜ : α < t ≤ β } . Given two distinct vectors x,y ∈ ℜ n , define [ x,y ] := { (1 − α ) x + αy : α ∈ [0 , 1] } , ( x,y ] := { (1 − α ) x + αy : α ∈ (0 , 1] } , [ x,y ) := { (1 − α ) x + αy : α ∈ [0 , 1) } , ( x,y ) := { (1 − α ) x + αy : α ∈ (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 ⊆ ℜ n is a subspace if it is closed under addition and scalar multiplication, that is, α x + α 1 x 1 ∈ V for every x ,x 1 ∈ V and α ,α 1 ∈ ℜ . exerc:sub Exercise 1.2.2 Show that a nonempty subset V ⊆ ℜ n is a subspace if, and only if, for every x,y ∈ V and α ≥ , the vectors x + y , − x and αx are in V . :affinemanifold Definition 1.2.3 A subset V ⊆ ℜ n is an affine manifold if αx +(1 − α ) x 1 ∈ V for every x ,x 1 ∈ V and α ∈ ℜ . In other words, a subset V of ℜ 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 ℜ 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 ℜ n is a subspace if, and only if, it is an affine manifold containing the origin. Exercise 1.2.4 Let V ⊆ ℜ 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 ⊆ ℜ n is the dimension of the subspace V − x , where x is an arbitrary point in V . Definition 1.2.6 A : ℜ n → ℜ m is an affine map if, for every x ,x 1 ∈ ℜ n and α ∈ ℜ : A ( αx + (1 − α ) x 1 ) = αA ( x ) + (1 − α ) A ( x 1 ) ....
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This note was uploaded on 08/24/2011 for the course ISYE 7682 taught by Professor Staff during the Fall '08 term at Georgia Tech.
 Fall '08
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