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From: jlg@lanl.ARPA
Newsgroups: net.math
Subject: Re: strange shapes
Message-ID: <18271@lanl.ARPA>
Date: Tue, 18-Dec-84 15:53:54 EST
Article-I.D.: lanl.18271
Posted: Tue Dec 18 15:53:54 1984
Date-Received: Thu, 20-Dec-84 05:35:58 EST
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Organization: Los Alamos National Laboratory
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> 
> 	To prove that a sphere is the 3-manifold of fixed area that 
> holds the largest volume is trivial. It is a simple iso-perimetric
> problem in the Calculus of Variations. See for example "Lectures on the
> Calculus of Variations", Bolza, Oskar, Dover Press, chapter 6.
> 
> 	It amounts to finding a function G(x,y, x', y') such that
> the double integral over G is fixed and for which the volume integral
> is maximal.

This assumes that the surface in question is continuously differentiable.
Or at least that the differential has a finite number of discontinuities.
If a solution to the stated problem exists, it maybe nowhere differentiable,
in which case the above mentioned proof is irrelevant.  (How do you compute
the area of a surface that is nowhere differentiable?  I suspect it depends
upon the method used to construct the surface.  I don't know.)

Actually, I don't think there is a finite surface that 'encloses' infinite
volume.  And I think that a sphere probably does enclose the maximal
volume for a given area.  But there have been no proofs yet.