Asymptotic estimates for best and stepwise approximation of convex bodies IV

Abstract

In this article we first prove a stability theorem for coverings in 𝔼² by congruent solid circles: if the density of such a covering is close to its lower bound , then most of the centers of the circles are arranged in almost regular hexagonal patterns. A version of this result then is extended to coverings by geodesic discs in two-dimensional Riemannian manifolds.

Given a sufficiently differentiable convex body C in 𝔼³, the following two problems are closely related: (i) Approximation of C with respect to the Hausdorff metric, the Banach-Mazur distance and a notion of distance due to Schneider by inscribed or circumscribed convex polytopes. (ii) Covering of the boundary of C by geodesic discs with respect to suitable Riemannian metrics.

The stability result for Riemannian manifolds and the relation between approximation and covering yield rather precise information on the form of best approximating inscribed convex polytopes P_n of C with respect to the Hausdorff metric: if the number n of vertices is large, then most of the vertices are arranged in almost regular hexagonal patterns. Consequently, the majority of facets of P_n are almost regular triangles. Here ‘regular’ is meant with respect to the Riemannian metric of the second fundamental form. Similar results hold for circumscribed polytopes and also for the Banach-Mazur distance and Schneider's notion of distance.