April 20th, 2016, 01:37 PM  #1 
Newbie Joined: Apr 2016 From: Pol Posts: 2 Thanks: 0  Faces of a convex set
I can't prove the following property of extreme subsets: Let A be a convex subset of $\displaystyle R^n$ 1) Prove that: G(x)={ $\displaystyle z \in$ A :$\displaystyle [xa(zx),x+a(zx)] \subset A$, for some $\displaystyle a>0$} Where G(x) is the intersection of the faces of A I have the following hints: This set is convex, If D is a convex subset of A and if F is a face of A such that ri(D) $\displaystyle \cap$ F is nonempty then D$\displaystyle \subset$F and for every x$\displaystyle \in$A if G is the intersection of the faces of A containing x, then x$\displaystyle \in$ri(G(x)). Where ri means relative interior. 
April 23rd, 2016, 03:44 AM  #2 
Senior Member Joined: Feb 2012 Posts: 144 Thanks: 16  
May 5th, 2016, 12:46 PM  #3 
Newbie Joined: Apr 2016 From: Pol Posts: 2 Thanks: 0 
G(x) is the intersection of the faces of A which containing x and x is random point which belongs to A.


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convex, faces, set 
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