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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


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?



