Lemma 1aThe perpendicular to

abatais a supporting line of A, and A is on the same side as B relative to that line.

Proof :

Let's namePthe perpendicular toabata. The fact thatPis a supporting line of A comes from the definition of Hausdorff distance. As illustrated below, ifais the furthest point of A relative tob, then a circle C centered atband of radiusabwill completely enclose A. Because C contains all points of A, then its tangent toais a supporting line of A.

The second part of lemma 1a says that A and B are on the same side of P. This is can be proved by contradiction : if some points of A are not on the same side of P than B, then the point of support (shown in green above) is not the furthest point of A from

b. This point of support thus can not define the Hausdorff distance h(A, B).