Wrapping Convex Bodies with Paper

This post has been updated on 27th November 2019

Recording some questions on optimally wrapping any given convex 3d body with a suitable single sheet of paper. This post is a prelude to further searches. First, a quote from a paper by Erik Demaine et al:

We consider three objectives in such wrappings: minimizing area of wrapper, minimizing perimeter, and the shape tiling the plane. Minimizing area naturally minimizes the material usage. Minimizing perimeter results in a minimum amount of cutting from a sheet of material....
Minimization of both area and perimeter (of the wrapper)... is a bicriterion optimization problem. Just minimizing the area is not interesting: starting from an arbitrarily thin rectangular strip, it is possible to wrap any surface using material arbitrarily close to the surface area.... However, Minimizing perimeter alone remains an interesting open problem.

With reference to the above, one feels that minimizing the area of the wrapper alone can become interesting if the paper is also constrained NOT to wind multiple times around the body being wrapped - this concept of 'winding' can be made precise, it seems. In what follows "least area" means "least area without multiple winding"

Another constraint that can be applied: the wrapping sheet ought to be a convex region. The least perimeter wrapping sheet for any convex 3d region is naturally convex. But is there any convex 3D region at all for which the least area wrapping sheet is convex? Seems the answer is NO.

I am not sure about the convex wrapper of least area and least perimeter for even a cube. To find a convex wrapper of any polyhedron, one can find a *net* of the polyhedron and take its convex hull. As given here , one can do better for a cube than to take a latin cross composed of 6 unit squares and find its hull (among the 11 nets for a cube, some have lower area convex hulls than the latin cross which has hull area = 9 units). Not sure if this is in general a good way to get a min area or min perimeter convex wrapper.

On the same note, what is the best least area(perimeter) wrapper (convex or otherwise) of a sphere? This page might give a clue. If convexity is not demanded, one can have a many pointed star wrapper whose area arbitrarily approaches that of the sphere itself. The convex hull of this wrapper is a circular region; if we use this disk as a wrapper, there will be plenty of wastage. But are there no more economical convex wrappers for the sphere?

One can ask for convex shapes for which the least area rectangular wrapper and the least perimeter rectangular wrapper are as different as possible.

And for what types of 'wrappees' will the least area and/or least perimeter convex wrappers of any given 3d body have smooth boundaries?

Another way to approach this problem area is to ask questions like: Given a square (or circular or whatever) sheet of paper, which is the max volume 3d body (convex or not necessarily convex) that can be wrapped by the sheet - crumpling and folding and gluing allowed but no tearing or stretching. Most such questions might be open, somewhat like the well-known paper bag or teabag problem.

Let me also link this discussion page: https://mathoverflow.net/questions/347122/trapping-3d-regions-with-sheets-of-paper