Inscribing an axially symmetric polygon and other approximation algorithms for planar convex sets

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&apos; that contains C. More precisely, for any epsilon > 0, we find an axially symmetric convex polygon Q subset of C with area vertical bar Q vertical bar > (1 - epsilon)vertical bar S vertical bar and we find an axially symmetric convex polygon Q&apos; containing C with area vertical bar Q&apos;vertical bar < (1 + epsilon)vertical bar S&apos;vertical bar. 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 l, find C boolean AND l. For instance, if C is a convex n-gon and its vertices are given in a sorted array, then T-C = O(logn). Then we can find Q and Q&apos; in time O(epsilon T--1/2(C) + epsilon(-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(epsilon T--1/2(C)).