Given two polygons, a dissection is a decomposition of the two polygons into a finite set of "nice" pieces with disjoint interiors, along with a bijection between the sets of pieces where a piece of A is matched to a congruent piece of B.
For example, here is Dudeney's famous dissection of an equilateral triangle into a square.
(One reasonable definition of "nice" is that each piece is a regular-closed set (aka equal to the closure of its interior) whose interior is connected and whose boundary is a Lebesgue null set.)
If there is a dissection between two polygons, is there a polygonal dissection between them with the same number of pieces? Furthermore, can we assume the polygonal dissection is arbitrarily close to the original, in some reasonable metric?
(Note - though I'm not sure why you would - that the reverse question is not true: Dudeney's dissection cannot be adjusted to a non-polygonal dissection. In fact, I think it's completely rigid - it cannot be adjusted to any other dissection at all.)
For a nontrivial example of a dissection between two polygons involving a curve, see this dissection between a Latin cross and a square using a circular arc.
