Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Posting-Version: version B 2.10.2 9/18/84; site lanl.ARPA Path: utzoo!watmath!clyde!burl!ulysses!unc!mcnc!philabs!cmcl2!lanl!jlg 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 References: <189@faron.UUCP> Sender: newsreader@lanl.ARPA Organization: Los Alamos National Laboratory Lines: 20 > > 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.