Optimal Bound on the Combinatorial Complexity of Approximating Polytopes

This article considers the question of how to succinctly approximate a multidimensional convex body by polytope. Given K unit diameter in Euclidean d -dimensional space (where is constant) and an error parameter ε > 0, objective determine polytope low combinatorial complexity whose Hausdorff distance from at most ε. By , we mean total number faces all dimensions. Classical constructions Dudley Bronshteyn/Ivanov show that O (1/ε ( -1)/2 ) facets or vertices are possible, respectively, but neither achieves both bounds simultaneously. In this article, it possible construct with complexity, which optimal worst case. Our result based on new relationship between ε-width caps its polar body. Using relationship, able obtain volume-sensitive bound approximating “essentially different.” We achieve our main combining variant witness-collector method novel variable-thickness layered construction economical cap covering.