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Ref: On folding a polygonal sheet, Multi-layered wrapping of polyhedra

Basic intent: to wrap a given convex planar lamina with a convex sheet of non-stretchable paper (such that every point on both sides of the lamina has at least one layer of paper covering it). To optimize, the wrapping convex paper sheet could be of least area/perimeter/diameter. An n-layered wrap is a wrap such that to reach any point on the lamina from outside, a 'needle' will have to pierce at least n layers of paper.

Question: Which is the unit area convex lamina for which the least area n-layered wrap needs a sheet of the largest area? For any n, is it always a unit disk?

Note: Same question can be asked for the least perimeter/diameter wrapping sheet.

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This paper doesn't directly answer your questions, but they do determine every convex polygon that can wrap a square.

Akiyama, Jin, and Koichi Hirata. "On convex developments of a doubly-covered square." In Indonesia-Japan Joint Conference on Combinatorial Geometry and Graph Theory, pp. 1-13. Berlin, Heidelberg: Springer Berlin Heidelberg, 2003.

Fig4

Of course the convex polygon wrapper has twice the area of the square.

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    $\begingroup$ Thank you. From another companion paper by Akiyama et al ('Flat 2-foldings of convex polygons') just learned that the convex region that 1-fold wraps (their 2-folding is a 1-fold wrap in the present post!) a unit square need not have a unit square as a subset. I couldn't quite figure out if they explicitly found out the least perimeter (or the max perimeter) convex polygon that 1-fold wraps a unit square! $\endgroup$
    – Nandakumar R
    Commented Jul 13, 2023 at 18:05
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    $\begingroup$ @NandakumarR: Those authors seemed not to be interested in the extreme conditions you posed: e.g., minimum perimeter. $\endgroup$
    – Joseph O'Rourke
    Commented Jul 13, 2023 at 22:27

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