0
$\begingroup$

what is the maxium area of a fixed perimeter closed graph?

can someone prove that the answer is a circle.

I can prove that the square have maxium area for a fixed perimeter rectangle.

  • 0
    A hexagon beats a square, and generally an $n$-gon gets better as $n$ increases, approaching a circle.2012-12-25
  • 2
    https://en.wikipedia.org/wiki/Isoperimetric_inequality2012-12-25
  • 0
    @coffeemath can we prove this rigorously?2012-12-25
  • 0
    That a hexagon beats a square is shown by a calculation. One has to find:[1] area for square of perimeter P, [2] area for regular hexagon of perimeter P. You may as well put P=1 to compare.2012-12-25
  • 0
    If you restrict to sufficiently smooth curves, proving the circle is optimal is a standard problem in the calculus of variations.2012-12-25

1 Answers 1