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Approximation of smooth convex bodies by circumscribed polytopes with respect to the surface area

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Abstract

Let K be a convex body with C 2 boundary in the Euclidean d-space. Following the work of L. Fejes Tóth, R. Vitale, R. Schneider, P.M. Gruber, S. Glasauer and M. Ludwig, best approximation of K by polytopes of restricted number of vertices or facets is well-understood if the approximation is with respect to the volume or the mean width. In this paper we consider the circumscribed polytope P (n) of n facets with minimal surface area, and present an asymptotic formula in terms of n for the difference of surface areas of P (n) and K.

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Authors and Affiliations

  1. Alfréd Rényi Institute of Mathematics, P.O. Box 127, Budapest, 1364, Hungary

    Károly J. Böröczky

  2. Department of Geometry, Roland Eötvös University, Pázmány Péter sétány 1/C, Budapest, 1117, Hungary

    Károly J. Böröczky & Balázs Csikós

Authors
  1. Károly J. Böröczky
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  2. Balázs Csikós
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Corresponding author

Correspondence to Károly J. Böröczky.

Additional information

Communicated by V. Cortés.

K.J. Böröczky supported by OTKA grants K 068398 and 075016, and by the EU Marie Curie TOK project DiscConvGeo.

B. Csikós supported by OTKA grants NK 067867 and K 072537.

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Böröczky, K.J., Csikós, B. Approximation of smooth convex bodies by circumscribed polytopes with respect to the surface area. Abh. Math. Semin. Univ. Hambg. 79, 229–264 (2009). https://doi.org/10.1007/s12188-009-0023-2

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  • DOI: https://doi.org/10.1007/s12188-009-0023-2

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