Hausdorff approximation of 3D convex polytopes

The algorithms involved in the Hausdorff approximation of 3D convex polytopes are discussed. The algorithms for 2D Hausdorff approximation are presented, which give precision almost equal to the best possible for the specific polygon involved. The constant involved in the estimate is strongly influenced by the desired degree of precision of the approximation. Hausdorff algorithm cannot be generalized to these dimensions, even by increasing the complexity, unlike the volume difference approximation. The algorithmic results are obtained, where a randomized algorithm is given which, for 3D polytopes, runs in time in the worst case. The convex polytopes can be approximated in the Hausdorff distance sense, by polytopes with fewer vertices inscribed in them, so that a best possible worst-case order of the error is achieved and the vertices of the approximating polytope constitute a subset of the vertices of the approximated polytope.