Given a square sheet of paper, how does one create a bag (a closed surface) with it such that the 3D region contained within this closed surface has maximum volume (operations allowed include wrinkling and sewing/gluing at the edges but not stretching or tearing)?

The closed surface need not be smooth. This question might be related to the 'paper bag (or teabag) problem'. Further, by varying the shape of the paper sheet (to say a circular or elliptical sheet), one has a range of questions.

Apart from solving each specific case, are there 'global' properties? For instance one could ask if these claims hold:

- "given any convex sheet, to produce a bag that can hold max volume, all sewing/gluing operations are done necessarily at the edges."
- "for any convex sheet, the bag formed with it holding maximum volume cannot be smooth when filled to capacity"