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April 20th, 2016, 01:37 PM   #1
Int
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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 :$\displaystyle [x-a(z-x),x+a(z-x)] \subset A$, 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.
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April 23rd, 2016, 03:44 AM   #2
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Quote:
Originally Posted by Int View Post
G(x) is the intersection of the faces of A.
what is x then?

besides, if A is bounded and its interior not empty, then the intersection of the faces of A is empty.
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May 5th, 2016, 12:46 PM   #3
Int
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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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