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Definitions (Inside-out polygonal dissections): a polygon P has an inside out dissection into P' if P′ is congruent to P and the perimeter of P becomes interior to P' and so the perimeter of P' is composed of internal cuts in the dissection of P. The dissection to P' is totally inside out (Further queries on inside-out polygonal dissections) if we further insist that no point on the boundary of P is on the boundary of P'. These definitions have natural analogs in 3D.

Let us drop the requirement that the final polygon (or solid) P' be congruent to P but only insist on the inside-out (or totally inside out) nature of the dissection of P into P'.

Question 1: Given a convex polygonal region, will it always have an inside out dissection into some convex polygonal region? If the answer is "yes" (as seems to be the case), is there some algorithm by which one achieve such a dissection for an n-gon with least number of intermediate pieces (we don't insist that these pieces are convex but are simple polygons)? If the answer is "no", how does one decide whether a given polygonal region has such a dissection?

Question 2: same question as 1 with the 'fully inside out' requirement.

Note added on 21st March, 2023: Following comment below and with reference to the Wallace Bolyai Gerwein theorem, one could guess that any polygon P can be inside-out dissected (or even fully inside out dissected) into any other polygon P' with the same area. So the existence part of both above questions seems to have a "yes" answer. What remains is given a P to find that P' into which it can be transformed via the least number of intermediate pieces.

Note: Some further thoughts are recorded at http://nandacumar.blogspot.com/2023/03/inside-out-dissections-contd.html

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    $\begingroup$ Isn't the answer to the 1st question in Q1 Yes because the original cited question showed that there is an inside-out dissection to a congruent $P'$? So loosening to a noncongruent $P'$ leaves the answer Yes? $\endgroup$
    – Joseph O'Rourke
    Commented Mar 19, 2023 at 0:12
  • $\begingroup$ Thanks very much for pointing this out. Indeed, one could triangulate P in such a way that each triangle has two of the edges of P and then inside-out each of these triangles - via 4 pieces each (as given in your original post). That easily gives an inside-out dissection of P. So the only issue with allowing P' to be any shape (probably) is whether P can be inside-outed into some shape other than itself via less intermediate pieces than into P itself. $\endgroup$
    – Nandakumar R
    Commented Mar 19, 2023 at 3:43
  • $\begingroup$ Taking things a little bit further, even for question 2 (totally inside out) the possibility answer appears to be "yes". With both questions, Specifying a specific P' could be of interest - whether one can inside out a general n-gon into a square, say. $\endgroup$
    – Nandakumar R
    Commented Mar 19, 2023 at 3:43

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