Approximation algorithms for inscribing or circumscribing an axially symmetric polygon to a convex polygon

Given a convex polygon P with n vertices, we present algorithms to determine approximations of the largest axially symmetric convex polygon S contained in P, and the smallest such polygon S′ that contains P. More precisely, for any ε > 0, we can find an axially symmetric convex polygon Q ⊂ P with area |Q| > (1 - ε)|S| in time O(n + 1/ε^3/2), and we can find an axially symmetric convex polygon Q′ containing P with area |Q′| < (1 + ε)|S′| in time O(n+(1/ε²)log(1/ε)). If the vertices of P are given in a sorted array, we can obtain the same results in time O((1/√ε) log n+1/ε^3/2) and O((1/ε) log n+(1/ε²) log(1/ε)) respectively.