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 this linked post).

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.