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This question is based on http://www.science.smith.edu/~jorourke/Papers/FoldingPP.pdf

Define an vessel as a convex polyhedron with one face removed - in other words, a vessel can be converted into a convex polyhedron by attaching to it exactly one convex polygonal face (the 'lid'). The capacity of a vessel is the volume of this convex polyhedron

Note: in a gravitational field, the vessel will have to be suitably oriented/mounted for it to be filled to its capacity; insisting on the stability of the filled vessel will give a variant to this question.

Question: Given a polygonal region, how does one fold/wrap it into a vessel of maximum capacity?

Remarks: For example, starting with a unit square, we can form (not sure of optimality) a 'tub' with dimensions (2/3 X 2/3 X 1/6 which is not uniformly thick but has more capacity than the volume of the (closed) largest closed polyhedron (of uniform thickness) that can be folded from the unit square (shown on 'Trapping' 3D regions with sheets of paper).

It seems that starting with a convex polygonal sheet, to achieve maximum capacity, one necessarily will have to allow for some overlap/ folding over of the material (thus leading to a vessel of non-uniform thickness) - a Greek cross can be folded into a cubical vessel without any overlap but it (the cross) is not convex.

Note added on 8th February, 2024: An alternative, looser definition of a 'vessel' would be "a connected subset of a closed convex surface in 3D", the interior of which can hold some liquid and the 'mouth' of which need not be a nice polygon (as in the defintion above).

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    $\begingroup$ A possible counterexample to the last claim (about necessarily having overlaps): start with an equilateral triangle, and cut three small equilateral triangles off at the corners. This folds without overlaps into a container which looks like a tetrahedron with the tip cut off. Can you get larger volume with overlaps? $\endgroup$
    – Achim Krause
    Jun 21, 2021 at 10:12
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    $\begingroup$ Another non-overlap example. Start with a pyramid whose apex has curvature at least $\pi$, say over a regular polygonal base. Remove the base. Slit one edge from the apex to a base vertex. Unroll onto the plane. One gets a convex fan (because of the curvature constraint). It seems like the baseless pyramid is the max volume vessel for this convex fan. $\endgroup$
    – Joseph O'Rourke
    Jun 21, 2021 at 10:52
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    $\begingroup$ Please note that the paper of mine you cite is from 2007, and the open problem mentioned at the end is answered: On Folding a Polygon to a Polyhedron. $\endgroup$
    – Joseph O'Rourke
    Jun 21, 2021 at 12:42
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    $\begingroup$ Thanks for those examples - I can see nothing wrong with them! - and for the pointer to the further folding paper. Hope the question of how to fold basic shapes such as the unit square and the greek cross into maximum capacity vessels retains some interest. Further, I understand that for folding convex polygons into polyhedrons of maximum volume, the max volume polyhedron is always a folding without overlaps - and for folding to vessels, this assertion does not appear to hold. $\endgroup$
    – Nandakumar R
    Jun 21, 2021 at 19:06

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