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Approximation of convex bodies by polytopes

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Abstract

LetC be a convex body ofE d and consider the symmetric difference metric. The distance ofC to its best approximating polytope having at mostn vertices is 0 (1/n 2/(d−1)) asn→∞. It is shown that this estimate cannot be improved for anyC of differentiability class two. These results complement analogous theorems for the Hausdorff metric. It is also shown that for both metrics the approximation properties of «most» convex bodies are rather irregular and that ford=2 «most» convex bodies have unique best approximating polygons with respect to both metrics.

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Authors and Affiliations

  1. Institut für Analysis, Technische Universität Wien, Gußhausstr. 27, A-1040, Vienna

    Peter M. Gruber & Petar Kenderov

  2. Institute of Math. and Mech, Bulgar. Acad. Sci., P. O. Box 373, BG-1090, Sofia

    Peter M. Gruber & Petar Kenderov

Authors
  1. Peter M. Gruber
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  2. Petar Kenderov
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Gruber, P.M., Kenderov, P. Approximation of convex bodies by polytopes. Rend. Circ. Mat. Palermo 31, 195–225 (1982). https://doi.org/10.1007/BF02844354

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  • DOI: https://doi.org/10.1007/BF02844354

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