Approximation algorithms for cutting a convex polyhedron out of a sphere

For a given convex polyhedron P of n vertices inside a sphere Q, we study the problem of cutting P out of Q by a sequence of plane cuts. The cost of a plane cut is the area of the intersection of the plane with Q, and the objective is to find a cutting sequence that minimizes the total cost. We present three approximation solutions to this problem: an O(nlogn) time O(log2n)-factor approximation, an O(n1.5logn) time O(logn)-factor approximation, and an O(1)-factor approximation with exponential running time. Our results significantly improve upon the previous O(n3) time O(log2n)-factor approximation solution.