In the proof of Lemma 1 we mentioned that many incorrect methods for finding a vertex p such that the line segment bp is an interior diagonal of P have been published. This exercise presents some of the incorrect ways p has been chosen in these proofs. Show, by considering one of the polygons drawn here, that for each of these choices of p, the line segment bp is not necessarily an interior diagonal of P.
a) p is the vertex of P such that the angle ∠abp is smallest.
b) p is the vertex of P with the least x-coordinate (other than b).
c) p is the vertex of P that is closest to b.
Exercises 22 and 23 present examples that show inductive loading can be used to prove results in computational geometry.