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Topic: Separation Hyperplane Theorem
Replies: 2   Last Post: Oct 12, 2012 3:20 PM

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Maury Barbato

Posts: 791
From: University Federico II of Naples
Registered: 3/15/05
Re: Separation Hyperplane Theorem
Posted: Oct 12, 2012 3:19 PM
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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 x-axis ( 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 half-spaces), 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 half-space determined by H.


Thank you very much for attention,adamk.
My Best Regards,
Maury Barbato



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