Constructing Strongly Convex Hulls Using Exact or

One useful generalization of the convex hull of a set S of n points is the-strongly convex-hull. It is deened to be a convex polygon P with vertices taken from S such that no point in S lies farther than outside P and such that even if the vertices of P are perturbed by as much as , P remains convex. It was an open question 1 as to whether an-strongly convex O()-hull existed for all positive. We give here an O(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-strongly convex O(+)-hull in O(n log n) time using rounded arithmetic with rounding unit. This is the rst rounded arithmetic convex hull algorithm which guarantees a convex output and which has error independent of n.