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Topic: A conjecture on convex symmetric curve. How to prove it?
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KaiJin

Posts: 1
From: beijing
Registered: 11/4/10
A conjecture on convex symmetric curve. How to prove it?
Posted: Nov 4, 2010 7:41 AM
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Suppose Q is a central symmetric convex planar curve. We define f(Q) to be max{Area(P)/Area(Q) | "P is a parallelogram inside Q."}

And we want to find f_min= min{f(Q)}.

It's trivial that f_min > 1/2. And with some exploration I can prove f_min> 4/(4+pi)=0.56 .
On the other hand, for any ellipse Q, f(Q)=2/pai. so f_min <=2/pai=0.6366 .

By notice that from any point on a ellipse we can draw a parrallelogram inside it whose area is global optimal (All equal to 2*a*b), it is natual to guess that f_min=2/pai.

Now I am quite sure of the correctness of this conjecture. But I don't know how to prove it!
I think it's very interesting. Could somebody help me?



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