This post continues from How big a box can you wrap with a given polygon? and Convex polyhedra that can be folded from convex polygons. One can also mention 'k-fold coverings of the plane' as examined in for example, 'Research problems in discrete geometry' by Brass, Moser and Pach.

Consider wrapping convex polyhedrons with polygonal regions with the additional requirement: the wrap should be at least $n$ layers thick - from any point on the surface of the polyhedron, to go out into space, we have to cut thru a minimum of $n$ layers of the wrap formed by the polygon. Note: Portions of the wrap can have more than $n$ lavers due to overlap of portions of the polygon.

  1. If a convex polyhedron cannot be wrapped (1-layer wrap) without overlaps by any convex polygon (eg. the cube), then, can it be 2-layer wrapped without overlaps by any convex polygon? The answer seems always negative.

  2. Are there convex polygons other than the rectangle with which we can wrap some convex polyhedron with 3 layers without overlap? Maybe the answer is negative.

  3. One can also ask: given a convex polygonal region, how big a sphere or box... can be 2-layer wrapped by it?

Note: A simpleminded approach to $n$-layer wrapping: First fold the wrapper polygon such that it is everywhere at least $n$ layers thick (how to do this folding to $n$-layers thickness seems nontrivial! A similar question was discussed at On folding a polygonal sheet) and then wrap the given polyhedron with this multilayered wrapper. But it might lead to portions of the wrap having up to $2n$ layers thickness (wastage) and hence may fail to wrap the largest possible object.



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