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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: (1t)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  
Global Moderator Joined: May 2007 Posts: 6,754 Thanks: 695  Quote:
 
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.


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closed, convex, convexity, find, subset 
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