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Earlier posts:

  1. How big a box can you wrap with a given polygon?
  2. Pairs of farthest points on surfaces of 3D convex solids

Ref with 'unfolding' defined: Agarwal, Pankaj K., Boris Aronov, Joseph O'Rourke, and Catherine A. Schevon. "Star unfolding of a polytope with applications." SIAM Journal on Computing 26, no. 6 (1997): 1689-1713.

Question: How does one unfold a polyhedron such that the convex hull of the unfolded embedding on the plane has the least area convex hull? Does the star unfolding have this property? If not, some other unfolding method?

Further question: The question in post 1 linked above can be turned inside out to seek the smallest area convex polygonal sheet of paper that can wrap a given convex polyhedron. Will ensuring the least area hull for the 'unfold' of the polyhedron achieve this?

Note: Same questions can be asked with area replaced with perimeter. But I am not sure if such a replacement could change the answers at least for some input convex polyhedrons.

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  • $\begingroup$ This in an interesting question but I think it warrants some clarification regarding the 'unfolding' notion. In the cited paper, they consider both the source unfolding and the star unfolding (both of which depend of the choice of a generic point $x$ in the polyhedron). In each case, it seems highly non-trivial that the unfolding admits an isometric embedding in the plane which corresponds to a simple polygon. Certainly this seems like a property you want so it seems to me this (and any other desired features) should be specified. $\endgroup$ Commented Jan 30 at 16:46
  • $\begingroup$ Thanks. I did mean an isometric embedding (distance preserving) because motivation came from wrapping a polyhedron with paper (can be bent and crumpled but not stretched). Beyond that if there are other considerations, I don’t know enough to comment. $\endgroup$ Commented Jan 31 at 9:21

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Not an answer. I think it's not straightforward to find an unfolding that minimizes hull area or perimeter. Even restricting to edge-unfoldings to nets seems difficult.

Here are the two min perimeter nets out of the $11$ cube nets.

CubeNets

Min perimeter nets play a role in self-folding polyhedra.

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  • $\begingroup$ Thanks… surprising that the question is so hard. It would be interesting to find some convex polyhedron that needs different unfoldings (even when unfoldings have to be of some restricted type) to achieve least hull area and least hull perimeter - in cube above, both unfoldings shown have same area and perimeter. $\endgroup$ Commented Jan 30 at 6:23
  • $\begingroup$ @NandakumarR: I shouldn't have said "there is little hope." Edited now. I don't know that it is hard. But there are many unfoldings to consider. $\endgroup$ Commented Jan 30 at 17:13
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    $\begingroup$ Thanks for explaining. It would be nice to know if for a general convex polyhedron, the four questions - its ‘ unfold’ with least area/perimeter and its ‘wrapping convex polygon’ with least area/perimeter - have different answers or if some of them are same. $\endgroup$ Commented Jan 31 at 9:29

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