Isoperimetric InequalitiesDate: 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 |
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