On wrapping solid bodies with planar regions

Given a sheet of paper P which is some planar region that cannot be stretched but can be folded or wrinkled at will. We want to find the 3d solid Q of largest volume that can be wrapped with P. We say "P wraps Q" if any line coming from far away has to pierce P at least once before it hits Q.

Can one think of general results (in Euclidean geometry) like this:

- whatever be the shape of P, the max volume Q it can wrap is necessarily non-convex.

- for any given convex 3d body, the P of least area that wraps it is necessarily non convex.

Can things be different in non-Euclidean geometry?

Remark:

I just saw that if the planar region is a long and very thin strip, the max volume 3d body it can wrap would be a sphere - just winding around it many times - which is convex. So, the question of finding least area wrapper for any 3d body might be meaningful only if we disallow such multiple windings - else a thin and v long strip with area equal to the surface area of the 3d body would be the answer.

However, the other question of finding the max volume a specified planar region can wrap around seems ok.