Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Posting-Version: version B 2.10.1 6/24/83; site faron.UUCP Path: utzoo!watmath!clyde!bonnie!akgua!whuxlm!harpo!decvax!linus!security!faron!bs From: bs@faron.UUCP (Robert D. Silverman) Newsgroups: net.math Subject: strange shapes Message-ID: <189@faron.UUCP> Date: Thu, 13-Dec-84 12:58:14 EST Article-I.D.: faron.189 Posted: Thu Dec 13 12:58:14 1984 Date-Received: Sun, 16-Dec-84 04:47:26 EST Organization: The MITRE Corp., Bedford, Ma. Lines: 10 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.