Bill T. writes: << . . . [T]here'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. . . . . . .
Let S^2 denote the unit sphere in R^3. The Nash embedding theorem -- as improved by Kuiper (cf. http://en.wikipedia.org/wiki/Nash_embedding_theorem ) -- implies that S^2 can be C^1 isometrically embedded into an arbitrarily small ball of R^3. QUESTION: Suppose we consider only isometric embeddings h_1 that arise from a continuous family of isometric embeddings h_t: S^2 -> R^3, 0 <= t <= 1, where h_0 is the inclusion mapping. Does there still exist an isometry h_1 of S^2 into an arbitrarily small ball in R^3? If not, how small can the image of such an isometry be? (It's not immediately obvious that the image can even have a smaller diameter than the original sphere, but it can.) --Dan