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Maury Barbato
Posts:
792
From:
University Federico II of Naples
Registered:
3/15/05


Re: Separation Hyperplane Theorem
Posted:
Oct 12, 2012 3:19 PM


adamk wrote:
> > Hi, > > let A and B two disjoint closed convex subsets of > > R^n. > > Then the standard separation hyperplane theorem > > states > > that there exists some linear functional such that > > L(x) <= L(y) for every x in A and y in B. I think > > that > > actually a stronger result holds, that is there > > exists > > a linear functional L such that L(x) < L(y) for > every > > > > x in A and y in B. What do you think about? > > Note that I have supposed that A and B are closed. > > Thank you very much in advance for your attention. > > My Best Regards, > > Maury Barbato > > How do you separate, e.g., the xaxis ( or : > {(x,y):y=0} > > and the line y=1/x , x>0 ( (x,y): y=1/x, x>0} ?
I think you meant the sets A={(x,y) : y=0} and B={(x,y) : y >= 1/x, x >0}. In this case the stright line y=0 is a separating line, or, in other words, if v=(0,1), then < v, z_1 > < < v, z_2 > for every z_1 in A and every z_2 in B. Note that I do not state that the two sets are strictly separated (in the usal meaning that they are contained in opposite open halfspaces), all that I require is that for some linear functional L, the strict inequality L(x) < L(y) holds. Geometrically this means that there exists a hyperplane H which not only separates the two sets, but such that one of the two sets is contained in the open halfspace determined by H.
Thank you very much for attention,adamk. My Best Regards, Maury Barbato



