Lenght of closed level curves
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From: LordBeotian on 11 May 2010 11:07 It seems reasonable to think that if we have a C^infty real function on the plane and a closed level curve C without singular points then there is another closed curve near C which has greater lenght. How could we build a proof?
From: Robert Israel on 11 May 2010 16:30
LordBeotian <pokipsy76(a)yahoo.it> writes: > It seems reasonable to think that if we have a C^infty real function > on the plane and a closed level curve C without singular points then > there is another closed curve near C which has greater lenght. How > could we build a proof? By "another closed curve" I assume you mean another closed level curve of the same function. No, this is false. Consider a function f(r,theta) (using polar coordinates) where f(0,theta) = 0, f(1,theta) = 1, df/dr > 0 for 0 < r < 1, and f(r,theta) = 1/2 on a curve inside the unit circle that has length > 2 pi. As a function of c in [0,1], the length of the level curve f(r,theta) = c must then have a local maximum somewhere in (0,1), and the level curve for such a value of c is a counterexample. -- Robert Israel israel(a)math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada |