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

but vertex 5 should be given preference because 9 is within the range given by 4-5 and 6-5

What would you do if 4-5 and 6-5 were even more convex so that 9 didn't lie within their range? Then by your rules the proper thing to do would be to connect 9 to 12 because 12 is the closest reflex vertex, which would be suboptimal.