CONSTRUCTING A STRONGLY CONVEX SUPERHULL OF POINTS

Abstract

Let S be a set of n points in the plane and CH(S) be the convex hull of S. We consider the problem of constructing an approximate convex hull which contains CH(S) with strong convexity. An ∊-convex δ-superhull of S is a convex polygon P satisfying the following conditions: (1) P has at most O(n) vertices, (2) P contains CH(S), (3) no vertex of P lies farther than δ outside CH(S), and (4) P remains convex even if its vertices are perturbed by as much as ∊. The parameters ∊ and δ represent the strength of convexity of P and the degree of approximation of P to CH(S), respectively. In this paper, we show that there exists an ∊-convex [Formula: see text]-superhull with at most n vertices for any S and any ∊ ≥ 0 and it can be constructed in O(n log n) time (in O(n) time is S is sorted).