Tom Karzes <karzes@sonic.net> wrote:

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.

Thanks. I mostly take a computational view of math, so that's not an issue for me. I'm interested in what my random-plane program should get as largest and smallest area and perimeter for each polygon. For instance the lower limits are obviously 0 for triangles. That doesn't mean I expect to find a triangle with that exact area or perimeter. It means that the more random planes I generate, the closer both numbers should get to 0. And indeed that's what I observe. It's only a problem if I do something stupid, such as take the geometric or harmonic mean, since those are dominated by values close to 0. I don't doubt that the intersection areas have well-defined geometric and harmonic means, and that they aren't zero. But they can't be found by a random plane program, not even with a googolplex planes. (They could be found with hairy triple integrals.) Fortunately, James Propp asked for the better-behaved arithmetic mean. I could also give him the quadratic mean (RMS) if he likes.