Maximal area in fixed perimeter An old story I heard starts by two people that was arguing about how much land a man need. So they called to a young man, and said to him:


  You are stating in that point, and start running all the day. All the area you cover before sunset will be yours. But you must return to the original point, otherwise you get nothing


So I considered what is the best route for him to get the maximal area.

More mathematical form:

Given a number d, what is the continues closed shape with the maximal area and perimeter d?

If we are limited to rectangles, the answer is "a square", what leads me to guess the optimal shape is a circle. Is it true and can it be proven?

What I can think about is that the shape must be convex, otherwise we can 'cut' it and save line. But I have no idea how to eliminate some strange elliptic shapes.
As @achille points out, the Isoperimetric inequality states that $$4\pi A\le L^2$$
Where $A$ denotes the area of a plane figure, and $L$ its perimeter.  Equality holds only in the case of a circle.

Another consequence of this is that for shapes of a given area, a circle has the smallest perimeter. (the dual of your statement).
It's not too hard to show the circle does better than the square for a given perimeter.

$ P_c = 2 \pi r $

$ P_{sq} = 4s $

The length of the run is equal in both cases

$2 \pi r = 4s $

$ r = \frac{2s}{\pi} $

$A_c = \pi r^2 = \pi (\frac{2s}{\pi})^2 = \frac{4s^2}{\pi} $

$ A_{sq} = s^2 $

Since $ \frac{4}{ \pi} &gt; 1 $ we see the circle encloses a greater area for a given perimeter to the square.