My Math Forum Faces of a convex set

 Real Analysis Real Analysis Math Forum

 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 :$\displaystyle [x-a(z-x),x+a(z-x)] \subset A$, 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.
April 23rd, 2016, 03:44 AM   #2
Senior Member

Joined: Feb 2012

Posts: 144
Thanks: 16

Quote:
 Originally Posted by Int 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.

 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.

 Tags convex, faces, set

 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post honzik Geometry 1 April 10th, 2015 12:58 AM sososasa Algebra 6 March 22nd, 2014 09:52 AM cr34m3 Computer Science 1 April 20th, 2010 07:59 AM frederico Real Analysis 0 April 6th, 2009 11:31 AM

 Contact - Home - Forums - Cryptocurrency Forum - Top