chapter  4
66 Pages

Convex Objects

Conversely, for an axis v for which the above inequality holds, we find that each plane H(v, 8) with max{v · y : y E B} < 8 < min{v · x : x E A} is a separating plane.

Suppose that A n B = 0. Then, v = a - b f: 0. Since a and b are closest points, vis the point closest to the origin of A-B, the CSO of A and B. Let x E A andy E B. Then, w = x -yEA -B. Since A -B is convex, anyu on the line segment vw is contained in A-B, and thus Hull~ llvll. It follows from Lemma 4.1 that Hvll 2 - v · w:::: 0. Hence, v · w ~ Hvll 2 > 0. We find that v · x > v · y for all x E A andy E B. Thus, vis a separating axis.