I haven't had a chance to look at this problem or the references, but there's a general principle, I think due to Nash (who proved a C^1 isometric embedding theorem) that was motivational at least implicitly if not explicit in some of Gromov's early work on partial differential inequalities, that any sub-isometric smooth embedding of a surface in space can be approximated by an isometric embedding. If you accept this principle, it's very easy to enclose a positive volume by gluing together two C^1 smooth disks which contract distances, then approximating by an isometric embedding. The idea of the principle is simple at least in concept: you start with smooth map that decreases length, then recursively put in small ripples or corrugations that increase lengths in one direction in one coordinate chart at a time to be close to the target value. These can be done on as fine a scale as you like, so if you make wrinkles on a scale where the surface looks planar, the ripples don't collide. The maximum change in slope is dependent only on the proportional increase of length you need, so the limit can be made C^1 smooth. Mylar has no trouble doing this in practice. C^2 smooth surfaces are of course a totally different story. Bill On Oct 11, 2007, at 10:39 AM, David Gale wrote:

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Here are some references for the corresponding problem using two unit disks identified along their boundaries:

The Mylar Balloon Revisited Ivaïlo M. Mladenov; John Oprea The American Mathematical Monthly, Vol. 110, No. 9. (Nov., 2003), pp. 761-784.

What Is the Shape of a Mylar Balloon? William H. Paulsen The American Mathematical Monthly, Vol. 101, No. 10. (Dec., 1994), pp. 953-958.

Both papers use some hairy integrals and differential geometry.

But, the first paper states that the goal is to determine the *convex hull* of the wrinkly shape that the Mylar balloon would actually assume. It's not clear to me what assumptions the second paper is making.

I doubt either paper rigorously determines the maximum volume the metric space formed by a circular teabag can assume when embedded isometrically in 3-space.

--Dan

It seems to me these papers do not require the embedding to be strictly an isometry.so I would still like to see an example, probably with "wrinkles", of the disc problem which has a positive volume.

dg

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David Gale Professor Emeritus Department of Mathematics University of California, Berkeley

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