Lemma 1bThe perpendicular to

abatbis a supporting line of B, andaand B are on different sides relative to that line.

Proof :

Lemma 1b is the obvious counterpart of lemma 1a, and will be proved using the same arguments. Ifbis the closest point of B froma, then a circle C of radiusabcentered atacontains only one point of B, namelyb. The tangent to C is thus a supporting line of B.

Similarly to the second part of lemma 1a, if some points of B are on the same side of P than A, then the support point is not the closest point of B from

a. This is opposed to the definition Hausdorff distance.