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Of all the curves that divide the circle into two regions with the same area, is the diameter the shortest one?

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An elementary proof is prefered! –  Sgernesto Dec 20 '12 at 9:44
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It can be shown, using a variational argument, that the curve should intersect the circle perpendicularly and has constant curvature. It's not hard to show that only the diameter is the only curve of this type dividing the disk in two equal halfs.

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Interesting. Never thought about it :) –  Gerenuk Dec 20 '12 at 11:21
    
@Martijn. Thanks for your comment. –  Sgernesto Dec 20 '12 at 12:27
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One has to add that all closed curves in the ball of radius $r$ not intersecting the circle have greater length than any diameter, since by the isoperimetric inequality they must have length at least $\sqrt{2} \pi r > 2 r$. –  Thomas Dec 20 '12 at 13:42
    
@Thomas. It remains the case where the curve has its ends in the circle line.As it's written below, this is clear if the the endpoints are opposite respect the center of the circle. –  Sgernesto Dec 20 '12 at 14:07
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