Asymptotic estimates for best and stepwise approximation of convex bodies - IV

In this article we first prove a stability theorem for coverings in E-2 by congruent solid circles: if the density of such a covering is close to its lower bound 2 pi/root 27 then most of the centers of the circles are arrang ed in almost regular hexagonal patterns. A version of this result then is e xtended to coverings by geodesic discs in two-dimensional Riemannian manifo lds. Given a sufficiently differentiable convex body C in E-3, the following two problems are closely related: (i) Approximation of C with respect to the H ausdorff 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 met rics. The stability result for Riemannian manifolds and the relation between appr oximation and covering yield rather precise information on the form of best approximating inscribed convex polytopes P-n of C with respect to the Haus dorff metric: if the number n of vertices is large, then most of the vertic es are arranged in almost regular hexagonal patterns. Consequently, the maj ority of facets of P-n are almost regular triangles. Here 'regular' is mean t with respect to the Riemannian metric of the second fundamental form. Sim ilar results hold for circumscribed polytopes and also for the Banach - Maz ur distance and Schneider's notion of distance.