Prove that (in Euclidean two-space) given a finite perimeter, the closed curve enclosing maximum area is a circle.
Before reading the rest of the solution, first read about Green's Theorem and area.
We begin with the area of a closed curve derived from Green's theorem and the formula for arclength:
Using Lagrange multipliers, we consider the expression which is like the action functional in classical mechanics.
If you want to learn about how Lagrange developed the multiplier method, click here.
Treating as the "Lagrangian", the Euler-Lagrange equation is or
Since is a scalar, we note that the above expression can be written which states that the curvature of this maximizing curve is constant. This would either be a circle or a straight line; however, straight lines are not closed and can possess infinite arclength. So given a finite perimeter, the circle encloses the maximum area.
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