Question: Given a convex polygonal region, how does one find the box (rectangular parallelopiped) of maximum volume that can be wrapped with this region? While wrapping, if needed, some portions of the polygon could overlap.
Possible variants to this question: replace box with say, sphere, cylinder or any other specific type of 3D object; replace volume with say, surface area.
Another Aspect: In https://erikdemaine.org/wrapping/, is stated: " if we can choose the piece of paper to be a thin rectangle, we can wrap a polyhedron with arbitrarily little wastage of paper: the amount of double coverage can be reduced to as close to zero as desired". Guess the definition of wrapping can be modified to exclude such 'winding wraps'.
Example: Once winding wraps are excluded, if we consider a given triangle and try to find the largest sphere that can be wrapped by it, where should should we place largest sphere on triangle and wrap? The incenter is a natural guess but am doubtful.
Adding at bit on July 18th, 2021:
One can further ask the general question about the highest volume/surface area general convex polyhedron that can be wrapped by a given polygon - how to find/characterize it in simple cases when the polygon is a triangle etc..
One can also ask which is the least area convex polygon with which say, a unit cube can be wrapped - without windings.
