Here are some references for the corresponding problem using two unit disks identified along their boundaries:
The Mylar Balloon Revisited IvaiÌlo 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