Decomposition to Convex Polygons

Question

This question is a little involved. I wrote an algorithm for breaking up a simple polygon into convex subpolygons, but now I\'m having trouble proving that it\'s not

Answer 1

I believe the regular five pointed star (e.g. with alternating points having collinear segments) is the counterexample you seek.

Edit in response to comments

In light of my revised understanding, a revised answer: try an acute five pointed star (e.g. one with arms sufficiently narrow that only the three points comprising the arm opposite the reflex point you are working on are within the range considered "good reflex points"). At least working through it on paper it appears to give more than the optimal. However, a final reading of your code has me wondering: what do you mean by "closest" (i.e. closest to what)?

Note

Even though my answer was accepted, it isn't the counter example we initially thought. As @Mark points out in the comments, it goes from four to five at exactly the same time as the optimal does.

Flip-flop, flip flop

On further reflection, I think I was right after all. The optimal bound of four can be retained in a acute star by simply assuring that one pair of arms have collinear edges. But the algorithm finds five, even with the patch up.

I get this:

removing dead ImageShack link

When the optimal is this:

removing dead ImageShack link