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 June 10th, 2016, 05:09 AM #1 Newbie   Joined: May 2016 From: Denmark Posts: 2 Thanks: 0 How to find out if the subset is closed? We have a vector p = (0, 0, 2) in R^3 and we have the subset S = {xp where x >= 0} + T, where T is the convex hull of 5 vectors: (2,2,2), (4,2,2), (2,4,2), (4,4,6) and (2,2,10). How do I show that the subset T is a closed and convex subset? I'm not sure if I'm correct, but I'm using the following definition of a subset being convex: A subset is called convex if it contains the line segment between any two of its points: (1-t)u + tv for every u and v in the subset. I've tried to take two of those 5 vectors and see if there contains a line segment, but so far, it doesn't make any sense. I hope that you can help me with this problem. June 10th, 2016, 02:43 PM   #2
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 Originally Posted by FightingMongooses We have a vector p = (0, 0, 2) in R^3 and we have the subset S = {xp where x >= 0} + T, where T is the convex hull of 5 vectors: (2,2,2), (4,2,2), (2,4,2), (4,4,6) and (2,2,10). How do I show that the subset T is a closed and convex subset? I'm not sure if I'm correct, but I'm using the following definition of a subset being convex: A subset is called convex if it contains the line segment between any two of its points: (1-t)u + tv for every u and v in the subset. I've tried to take two of those 5 vectors and see if there contains a line segment, but so far, it doesn't make any sense. I hope that you can help me with this problem.
A convex hull is, by definition, convex. June 26th, 2016, 06:15 AM #3 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 The "convex hull" of a set of points (or vectors defining those points) is defined as "the smallest convex set containing those points" so, as mathman said, a "convex hull" is convex by definition. The simplest way to prove it is closed is probably to show that it contains all of its boundary points- and the boundary points of a convex hull of a (finite) set of points always consists of points on a straight line between two of the given points. Tags closed, convex, convexity, find, subset Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post nicnicman Linear Algebra 4 November 25th, 2013 08:38 AM 450081592 Real Analysis 2 October 31st, 2011 06:02 PM kickup Linear Algebra 2 November 8th, 2010 01:16 PM mapleleafs89 Applied Math 12 October 20th, 2009 02:59 PM Patricia-donnelly Real Analysis 3 April 9th, 2008 07:01 AM

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