Constructing strongly convex hulls using exact or rounded arithmetic

One useful generalization of the convex hull of a set S of points is the ε-strongly convex δ-hull. It is defined to be a convex polygon Rgr; with vertices taken from S such that no point in S lies farther than δ outside Rgr; and such that even if the vertices of Rgr; are perturbed by as much as ε, Rgr; remains convex. It was an open question as to whether an ε-strongly convex Ο(ε)-hull existed for all positive ε. We give here an Ο(n log n) algorithm for constructing it (which thus proves its existence). This algorithm uses exact rational arithmetic. We also show how to construct an ε-strongly convex Ο(ε + μ)-hull in Ο(n log n) time using rounded arithmetic with rounding unit μ. This is the first rounded arithmetic convex hull algorithm which guarantees a convex output and which has error independent of n.