Imagine the surface made by rotating a 'simple' strict convex function f(x) (x in [a, b] and by 'simple', I suppose f(a)=max(f(x)), f(b)=min(f(x)), around x=a, we get something like a peak of a montain. I found here is something quite related: http://www.chass.utoronto.ca/~osborne/MathTutorial/QCCF.HTM
Then let us put several such peaks in the plane and they may overlap with each other, then the height of the overlapped areas are the sum of all the peaks that overlap. So i wonder for the resulting surface, do we still have the similiar propoter of the convex function with one variable, for example, the maximum occurs only at the peak points?
I have the following concerning of the single variable convex functions: if we have a group of stictly convex function f_k(x), x in [a,b], then the sum them( or any linear combination of them) is still covex, isn't it? If this is true, we can say the maximum of the sum occurs at either of the two end points( or both when they are the same), isn't it? Can we extend this to a two or multi variable case?
Thank you for any comments, any avialable results or any references.
"Bing" <hanbin...@gmail.com> wrote: > Imagine the surface made by rotating a 'simple' strict > convex function f(x) (x in [a, b] and by 'simple', I suppose > f(a)=max(f(x)), f(b)=min(f(x)), around x=a, we get something like a > peak of a montain. I found here is something quite related: > http://www.chass.utoronto.ca/~osborne/MathTutorial/QCCF.HTM
> Then let us put several such peaks in the plane and they may overlap > with each other, then the height of the overlapped areas are the sum > of all the peaks that overlap. So i wonder for the resulting surface, > do we still have the similiar propoter of the convex function with one > variable, for example, the maximum occurs only at the peak points?
> I have the following concerning of the single variable convex > functions: if we have a group of stictly convex function f_k(x), x in > [a,b], then the sum them( or any linear combination of them) is still > covex, isn't it? If this is true, we can say the maximum of the sum > occurs at either of the two end points( or both when they are the > same), isn't it? Can we extend this to a two or multi variable > case?
Not sure this speaks to your question, but sin x has its maximum (on [0, pi/2]) at pi/2 and cos x has its maximum at 0 while sin x + cos x has its maximum at the interior point pi/4.
What's a propoter?
-- Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email)
> In article <1173215957.395012.278...@s48g2000cws.googlegroups.com>,
> "Bing" <hanbin...@gmail.com> wrote: > > Imagine the surface made by rotating a 'simple' strict > > convex function f(x) (x in [a, b] and by 'simple', I suppose > > f(a)=max(f(x)), f(b)=min(f(x)), around x=a, we get something like a > > peak of a montain. I found here is something quite related: > >http://www.chass.utoronto.ca/~osborne/MathTutorial/QCCF.HTM
> > Then let us put several such peaks in the plane and they may overlap > > with each other, then the height of the overlapped areas are the sum > > of all the peaks that overlap. So i wonder for the resulting surface, > > do we still have the similiar propoter of the convex function with one > > variable, for example, the maximum occurs only at the peak points?
> > I have the following concerning of the single variable convex > > functions: if we have a group of stictly convex function f_k(x), x in > > [a,b], then the sum them( or any linear combination of them) is still > > covex, isn't it? If this is true, we can say the maximum of the sum > > occurs at either of the two end points( or both when they are the > > same), isn't it? Can we extend this to a two or multi variable > > case?
> Not sure this speaks to your question, > but sin x has its maximum (on [0, pi/2]) at pi/2 > and cos x has its maximum at 0 > while sin x + cos x has its maximum at the interior point pi/4.
> What's a propoter?
> -- > Gerry Myerson (g...@maths.mq.edi.ai) (i -> u for email)
But note that sin and cos functions are concave on [0 pi/2], not convex. They are convex on [pi, 3*pi/2] so the maximum of sin x + cos x are at pi and 3*pi/2 with zero.
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