Lemma 3Let b

_{i}be the closest point of B from a vertex a_{i }of A. If µ is the moving direction (clockwise or counterclockwise) from b_{i }to b_{i+1}then, for a complete cycle through all vertices of A, µ changes no more than twice.

Proof :

Let's first illustrate this lemma with an example :As shown above, moving from b

_{1}to b_{2}follows a counterclockwise path on the boundary of B, while b_{3}to b_{5}implies a clockwise traversal. Because no movement occured from b_{2}to b_{3}, b_{3}is called astationarypoint.So, suppose now we go along a chain monotone in some direction

d, from a point a_{n}to a_{n+1}:If we put by hypothesis that b

_{n}is the closest point of B from a_{n}, where can be located b_{n+1}? We already proved (lemma 1b) that line L_{ }is a supporting line of polygon B, which is located below L. So b_{n+1}can only be below L.

However, b_{n+1}can not be below L at the right of P : any point in that area is further from a_{n+1}than b_{n}. If there is no point of B on the left side of P, then b_{n+1}could only be at the same place than b_{n}. Except for this case, b_{n+1}can not be neither on P, because any point on P below L is also further from a_{n+1}than b_{n}.

So b_{n+1}is necessarily below L, but most important, at the left of P, that is in the same directiondthat we went from a_{n}to a_{n+1}. We conclude that moving along a directiondon a monotone chain of a_{i}'s make b_{i}'s move in the same direction. In the worst case, some b_{i}may be at the same place than b_{i-1}(stationary point), but this doesn't change our conclusion.

Proving lemma 3 is now straightforward :_{ }travelling around a convex polygon is like moving on two monotone chains of opposite directions. Because b_{i}'s will move in the same direction than a_{ i}'s (or maybe not move at all), then the moving direction of b_{i}'s can not change more than twice.Q.E.D.

Obviously, lemma 3 doesn't say that µ

willchange twice : for instance, if the starting vertex of the figure above would have been the leftmost or rightmost point of A, then only one change of direction would have had occured. Similarly, we can easily find polygons A and B such that there would be no change in µ, because b_{i}'s would move only in one direction, or not move at all.