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.

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.

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.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.