Wrapping a 2D lamina with paper

Basic question: to wrap a given planar region with a convex sheet (such that every point on both sides of the lamina has at least one layer of paper covering it) with the wrapping convex sheet being of least area/perimeter. An n-fold wrap is a wrap such that to reach any point on the lamina from outside, a 'needle' will have to cut through at least n layers of paper.

It is not difficult to cook up convex 2D regions such that the least area wrapper and least perimeter wrapper are different. Here is a pentagon for which the least area wrapper is a 7-gon and the least perimeter wrapper appears to be a rectangle.

Note that above pic is a rough one. The x length of the pentagon being wrapped ought to be considerably more than the y height (say, twice). the slope of the near horizontal tilted edges should be much less than the pic indicates. Indeed, the pentagonal least area wrapper is only very marginally different from a rectangle.
Overall, the 7-gon wrapper (least area) is almost a rectangle with dimensions say 20 X 5 and the rectangular wrapper (least perimeter) has dimensions 10X11 approximately.

Questions:
- Which planar convex region of unit area has the least area wrapper having max area? Is it a disk?
- Which planar convex region of unit perimeter has least perimeter wrapper with max perimeter?
- Same questions as above with the wrapping generalized to an n-fold one.