Two constrained versions of the main question given in this post: Polyhedrons with mutually non-congruent faces, all of equal area. An earlier post that could be related: Cutting a spherical surface into mutually non-congruent pieces of equal area
for any integer $n>4$, are there $n$-hedra inscribed in a sphere (all vertices lying on the spherical surface) such that the faces are all of equal area and mutually non-congruent?
If one asks the above question about $n$-hedra with equal area and mutually non-congruent faces such that a sphere touches all its faces internally, is that a qualitative change?
Note: The questions can be asked with "area" replaced by "perimeter", "diameter" and so forth. And I don't know if replacing the "sphere" with some other closed convex surface would have implications.
