Access the full text.
Sign up today, get DeepDyve free for 14 days.
References for this paper are not available at this time. We will be adding them shortly, thank you for your patience.
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 , 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 𝔼 3 , 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.
Forum Mathematicum – de Gruyter
Published: Nov 1, 1998
Read and print from thousands of top scholarly journals.
Already have an account? Log in
Bookmark this article. You can see your Bookmarks on your DeepDyve Library.
To save an article, log in first, or sign up for a DeepDyve account if you don’t already have one.
Copy and paste the desired citation format or use the link below to download a file formatted for EndNote
Access the full text.
Sign up today, get DeepDyve free for 14 days.
All DeepDyve websites use cookies to improve your online experience. They were placed on your computer when you launched this website. You can change your cookie settings through your browser.