some problem connected with prove homogeneous inequalities
Pawel_Iks
interested in true inequalities homogeneous of n-th order in variables
x,y,z, ... The problem is to find all such inequalities.
1) for n=1, we have:
A*(x+y)>=0, so the constant A is from S(A)={A: A>=0}
2) for n=2, we have:
A*(x^2+y^2)+B*x*y>=0, it's easy to show that constants A,B
have to satisfy inequality
abs(B)>2*abs(A), so S(A,B)={(A,B): abs(A)>2*abs(B)}, and this
set is a 2D cone with origin at (0,0)
3) for n=3, we have
A*(x^3+y^3+z^3)+B*(x^2y+y^*x+...)+C*xyz>=0,
I suppose that S(A,B,C) is also a cone with origin at (0,0,0), but I
have no idea how to prove it
4) for n=4, we have:
A(x^4+y^4+z^4+t^4)+B(xyz^2+...)+C(x^2y^2+...)+D(x^3y+...)+Exyzt>=0
I'm interesting if it's also a cone S(A,B,C,D,E) ...
and if it's a general rule. Are there some simply methods for finding
such sets?
Robert Israel
Of course: for any set S, the functions from S to [0, infty) form
a cone. That is, if f_1 and f_2 are such functions and t_1, t_2
are in [0,infty), then t_1 f_1 + t_2 f_2 is such a function.
> and if it's a general rule. Are there some simply methods for finding
> such sets?
--
Robert Israel isr...@math.MyUniversitysInitials.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
Robert Israel
>
> > Let's consider real x,y,z,... from range (0,infinity). We are
> > interested in true inequalities homogeneous of n-th order in variables
> > x,y,z, ... The problem is to find all such inequalities.
> >
> > 1) for n=1, we have:
> > A*(x+y)>=0, so the constant A is from S(A)={A: A>=0}
Why not A*x + B*y?
> > 2) for n=2, we have:
> > A*(x^2+y^2)+B*x*y>=0, it's easy to show that constants A,B
> > have to satisfy inequality
> > abs(B)>2*abs(A), so S(A,B)={(A,B): abs(A)>2*abs(B)}, and this
> > set is a 2D cone with origin at (0,0)
Are you assuming your function is not just homogeneous but also symmetric in
the variables? How does the number of variables relate to n?
Pawel_Iks
> the variables?
Yes, of course. I've forgotten to mention it :(
> How does the number of variables relate to n?
I don't think about it, however it could be computed in some
combinatorial way, and I think that it isn't difficult. However, I'm
interesting not in it, but how to compute the bounds of the set S (for
n=2 we have the intersection of two half planes: y>=+-2x.
Pawel_Iks
> > Are you assuming your function is not just homogeneous but also symmetric in
> > the variables?
>
> Yes, of course. I've forgotten to mention it :(
>
> > How does the number of variables relate to n?
>
i didn't note that you were asking about number of variables (I though
that you are asking about constants A,B,...), for given number n,
there are n variables, becouse I want to have term x_1*x_2*...*x_n.