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We add one more bit to Forming paper bags that can 'trap' 3D regions of max surface area (note: some possibly open related questions are also in the comments following the answer to above linked post and the reference given at the end).

As was stated in the answer to above question: given a planar convex region S, one can perform perimeter halving operations on it and achieve 3D convex regions from the halved region. If S is polygonal, one can form infinitely many different convex polyhedra by different halvings of S.

Question: Given a convex polygonal region S, how does one maximize the volume of the resultant polyhedron? How does one maximize the width (least distance between a pair of parallel planes that touch the polyhedron) of the polyhedron? Will the same polyhedron satisfy both?

These questions might be related to the issue (mentioned in above linked post) of characterizing those polyhedrons which could be got via perimeter halving of convex polygons.

Ref: https://www.science.smith.edu/%7Ejorourke/Papers/FoldingPP.pdf

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This is only a remark about one polygonal region $S$, a square.

As mentioned in this posting, I computed numerically the max volume shape that can be made from any folding of a square. Here it is:

MaxVol

The red path is the boundary of the square, perimeter-halved and joined as shown. Figure from Geometric Folding Algorithms: Linkages, Origami, Polyhedra.

I have no insight into what makes this the max volume shape, and I suspect this is a difficult problem for arbitrary $S$.

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  • $\begingroup$ Thanks. Even this special case looks so complex. Although I am unaware of them, there ought to be other methods to make convex polyhedrons out of convex planar regions. And it might well be the case that, in most cases, to maximize the volume of the polyhedron some part off the initial polygon's area ought to be 'wasted' - unlike in perimeter halving. $\endgroup$
    – Nandakumar R
    Commented Feb 7 at 7:19

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