A Simple Proof of an Estimate for the Approximation of the Euclidean Ball and the Delone Triangulation Numbers

✍ Scribed by Piotr Mankiewicz; Carsten Schütt

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 125 KB
• Volume: 107
• Category: Article
• ISSN: 0021-9045
• DOI: 10.1006/jath.2000.3509

✦ Synopsis

We give a simple proof of an estimate for the approximation of the Euclidean ball by a polytope with a given number of vertices with respect to the volume of the symmetric difference metric and relatively precise estimate for the Delone triangulation numbers. We also study the same problem for a given number of n&1-dimensional faces.

2000 Academic Press

In this note we present a simple proof of an estimate for the approximation of a convex body by a polytope due to Gordon, Reisner, and Schu tt [GRS].

By B n 2 we denote the Euclidean ball in R n . Recall that the Hausdorff distance between two convex bodies K and C is defined by

where & } & denotes the usual Euclidean norm on R n , and that the symmetric difference metric is the volume of the symmeric difference of K and C, d S (K, C)=vol n (K q C).