Abstract

Abstract Given a planar convex set C, we give sublinear approximation algorithms to determine approximations of the largest axially symmetric convex set S contained in C, and the smallest such set S that contains C. More precisely, for any ε > 0, we find an axially symmetric convex polygon Q ⊂ C with area |Q| > (1 − ε)|S| and we find an axially symmetric convex polygon Q containing C with area |Q | < (1 + ε)|S |. We assume that C is given in a data structure that allows to answer the following two types of query in time T C : given a direction u, find an extreme point of C in direction u, and given a line , find C ∩ . For instance, if C is a convex n-gon and its vertices are given in a sorted array, then T C = O(log n). Then we can find Q and Q in time O(ε −1/2 T C + ε −3/2 ). Using these techniques, we can also find approximations to the perimeter, area, diameter, width, smallest enclosing rectangle and smallest enclosing circle of C in time O(ε −1/2 T C ).  2005 Elsevier B.V. All rights reserved.