Subject: Isoperimetric property of regular n-gons
Date: Thu 12/17/98 9:05 AM
From: Himanshu Roy Pota

"of all equilateral and equiangular plane figures having the same perimeter, that which has the greater number of angles is always greater, and the greatest of them all is the circle having its perimeter equal to them."

In your article on Isoperimentric theorem you make the above statement. I am very keen to have a proof of this statement. As a matter of fact I have wasted several hours trying to prove a similar thing. I have been trying to prove this following these steps:

  1. Given an isoscles triangle it can be transformed to a quadrilateral with the same base as the triangle, same perimeter but equal or larger enclosed area.
  2. This means that every n-gon can be transformed to n+1-gon with the same perimenter.
  3. Then we can use the fact that for any n-gon the largest enclosed area for a given perimeter is for a regular n-gon.
  4. In the limit n-gon polygon will be a circle.

Help me to prove 1 if you can or send me the proof you have.

Thanks for your attention.

Himanshu

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