I think Fred's point is that the maximum (or minimum) may be a limit case, where the limit polygon is not a pentagon, even though the polygons leading to the limit are. Tom Keith F. Lynch writes:

Fred Lunnon <fred.lunnon@gmail.com> wrote:

...

Do you even know that a pentagonal maximum exists --- the supremum might occur at some maximal quadrilateral!

Huh? There are intersections between the plane and the cube that have 3, 4, 5, and 6 sides. For each number of sides, there's a smallest area and a largest area. And a smallest perimeter and a longest perimeter.

Speaking of which, I should also ask what the smallest values should be for 5 and 6 sides. (For 3 and 4 sides, the minimum areas are obviously 0 or infinitesimal. For 3 sides the shortest perimeter is 0 or infinitesimal, and for four sides it's 4, or infinitesimally more.)

Again, I'm speaking of planes in all possible orientations and locations, and a cube with a side length of 2. (Why not a unit cube? Because I can visualize larger cubes better! :-) )