Plane Figure with Maximum Area for given Perimeter is Circle

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Theorem

Let $F$ be a plane figure.

Let $P$ be the length of the perimeter of $F$.

Let the area of $F$ be the largest of all the plane figures whose perimeters are of length $P$.


Then $F$ is a circle.


Poof


Historical Note

The problem of determining the plane figure which has the maximum area for a given perimeter was first solved by Jacob Bernoulli.

He also went ahead with a generalisation of the problem.


Sources