Approximation by Convex Polytopes

After some general introductory remarks on approximation in convex geometry we present in the main part of this article asymptotic results on best approximation of convex bodies as the number of vertices, resp. facets of the approximating poly topes tends to infinity. Tools are from (affine) differential geometry. Since the transparent geometric situation in the plane admits much more precise results, this case is treated separately. In higher dimensions the relations to Dirichlet- Voronoi and Delone tilings and to the ball covering problem are indicated. Then algorithmic and, in particular, step-by-step approximation results are discussed. Here tools from number theory are applied. In the material presented emphasis is on the ideas underlying the proofs. Supplementing earlier surveys, the last chapter contains a summary of recent results on approximation of convex bodies.

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