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