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

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 C ^T: given a direction u, find an extreme point of C in direction u, and given a line ℓ, find C∩ℓ. For instance, if C is a convex n-gon and its vertices are given in a sorted array, then C ^T=O(logn). Then we can find Q and Q′ in time O(ε ^-1/2C ^T+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 ^-1/2C ^T).