# L3_sets_geom - Lecture 3 Convex Set Topology and Operations...

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Lecture 3: Convex Set Topology and Operations Preserving Convexity January 24, 2007

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Lecture 3 Outline Review basic topology in R n Open Set and Interior Closed Set and Closure Dual Cone Recession Directions of a Convex Set Lineality Space Convex Set Decomposition Operations Preserving Convexity Affine Transformation Perspective Transformation Convex Optimization 1
Lecture 3 Topology Review Let { x k } be a sequence of vectors in R n Def. The sequence { x k } ⊆ R n converges to a vector ˆ x R n when x k - ˆ x tends to 0 as k → ∞ Notation: When { x k } converges to a vector ˆ x , we write x k ˆ x The sequence { x k } converges ˆ x R n if and only if for each component i : the i -th components of x k converge to the i -th component of ˆ x | x i k - ˆ x i | tends to 0 as k → ∞ Convex Optimization 2

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Lecture 3 Open Set and Interior Let X R n be a nonempty set Def. The set X is open if for every x X there is an open ball B ( x, r ) that entirely lies in the set X , i.e., for each x X there is r > 0 s.th. for all z with z - x < r, we have z X Def. A vector x 0 is an interior point of the set X , if there is a ball B ( x 0 , r ) contained entirely in the set X Def. The interior of the set X is the set of all interior points of X , denoted by int ( X ) How is int ( X ) related to X ? Example X = { x R 2 | x 1 0 , x 2 > 0 } int ( X ) = { x R 2 | x 1 > 0 , x 2 > 0 } int ( S ) of a probability simplex S = { x R n | x 0 , e x = 1 } Th. For a convex set X , the interior int ( X ) is also convex Convex Optimization 3
Lecture 3 Closed Set Def. The complement of a given set X R n is the set of all vectors that do not belong to X : the complement of

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