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Optimal growth order of the number of vertices and facets in the class of Hausdorff methods for polyhedral approximation of convex bodies

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Abstract

The internal polyhedral approximation of convex compact bodies with twice continuously differentiable boundaries and positive principal curvatures is considered. The growth of the number of facets in the class of Hausdorff adaptive methods of internal polyhedral approximation that are asymptotically optimal in the growth order of the number of vertices in approximating polytopes is studied. It is shown that the growth order of the number of facets is optimal together with the order growth of the number of vertices. Explicit expressions for the constants in the corresponding bounds are obtained.

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Authors and Affiliations

  1. Universidad Rey Juan Carlos, Mostoles, Madrid, 28933, Spain

    R. V. Efremov

  2. Dorodnicyn Computing Center, Russian Academy of Sciences, ul. Vavilova 40, Moscow, 119333, Russia

    G. K. Kamenev

Authors
  1. R. V. Efremov
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  2. G. K. Kamenev
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Correspondence to G. K. Kamenev.

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Original Russian Text © R.V. Efremov, G.K. Kamenev, 2011, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2011, Vol. 51, No. 6, pp. 1018–1031.

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Efremov, R.V., Kamenev, G.K. Optimal growth order of the number of vertices and facets in the class of Hausdorff methods for polyhedral approximation of convex bodies. Comput. Math. and Math. Phys. 51, 952–964 (2011). https://doi.org/10.1134/S0965542511060054

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  • DOI: https://doi.org/10.1134/S0965542511060054

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