2
$\begingroup$

What is a direct proof of isoperimetric inequality? In another word, i am asking for a proof that the circle has the maximum area compare to other geometric shape with the same circumstance but without the assumption that the circle is the "optimal choice".

Could anybody answer?

Comment 1: i think the circle S1 absolutly comes from nowhere, then immediatialy substitude some type of inequality...

Here is the elemetary indirect proof from Do Carmo's Differential Geometry:

Page 1 of the indirect proof: http://i.stack.imgur.com/UXDnD.png

page 2: http://i.stack.imgur.com/FuHnA.png

Page 3: http://i.stack.imgur.com/EVsXc.png

  • 1
    What do you mean by "the assumption that the circle is the 'optimal choice'"? That's generally the conclusion of the isoperimetric inequality, not its hypothesis.2012-10-12
  • 0
    @OwenBiesel - i think the circle S1 absolutly comes from nowhere, then immediatialy substitude some type of inequality...2012-10-12
  • 0
    What do you mean by "$S^1$ absolutely comes from nowhere"?2012-10-12
  • 0
    @neal - I assumed that no mathematician define the circle so far, how could one discover this inequality?2012-10-12
  • 0
    I'm still not sure what you mean. You can't conclude that the circle is optimal without defining the circle.2012-10-13

0 Answers 0