Approximating Convex Polyhedra with Axis-Parallel Boxes

Abstract
In this paper, we present an O(n4log2n) time algorithm to compute an approximate discrete axis-parallel box of a given n-vertex convex polyhedron P such that the given polyhedron is minimized. Here, "discrete" means that each plane containing a face of the approximate box passes through a vertex of P (or, more generally, passes through a point of a set of given points). This algorithm is significantly faster than the brute force O(n7) time solution for computing the optimal approximate axis-parallel box A* of P such that the symmetric difference of the volume between P and A* is minimized. We present a linear time algorithm to compute a pseudo-optimal (with factor approximate axis-parallel box of a convex polyhedron under the Hausdorff distance criterion. We also present O(n) and O(n7logn) time algorithms to compute the optimal approximate ball, with or without a fixed center, of a convex polyhedron under the Hausdorff distance criterion.

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