Isoperimetric Inequalities

Date: 12/16/95 at 12:14:58
From: Anonymous
Subject: Geo

How can I prove that a circle has more area inside given its 
border length than any other shape?

Date: 5/30/96 at 11:23:53
From: Doctor Tom
Subject: Re: Geo

This is known as the "Isoperimetric Inequality," and is a standard
theorem in differential geometry or minimal surfaces. The proof
is not at all easy. If you understand advanced calculus, you can
probably read a proof in a book on differential geometry.

For example, on page 33 of "Differential Geometry of Curves and
Surfaces," by Manfredo Do Carmo, there is a proof of the
isoperimetric inequality for the plane (your problem). The proof
takes two full pages, and depends heavily on a complete 
understanding of the material before it in the book.

I have never seen a simple proof of this important theorem.

The lesson to be learned is that in mathematics, it is often
easy to state a problem that is very difficult or impossible
to prove.

-Doctor Tom,  The Math Forum