Abstract
We prove that roughly \(\frac{d}{(1-\vartheta )^d}\ln \frac{1}{(1-\vartheta )^d}\) points chosen uniformly and independently from a centered convex body K in \({\mathbb {R}}^d\) yield a polytope P for which \(\vartheta K\subseteq P\subseteq K\) holds with large probability. This gives a joint generalization of results of Brazitikos, Chasapis and Hioni and of Giannopoulos and Milman.
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Acknowledgements
The author thanks Nabil Mustafa for enlightening conversations on the \(\varepsilon \)-net theorem and topics around it. I also thank the anonymous referees whose remarks helped fix some errors and improve the presentation. The research was partially supported by the National Research, Development and Innovation Fund Grant K119670, and by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences, and by the ÚNKP-17-4 New National Excellence Program of the Ministry of Human Capacities. Part of the work was carried out during a stay at EPFL, Lausanne at János Pach’s Chair of Discrete and Computational Geometry supported by the Swiss National Science Foundation Grants 200020-162884 and 200021-165977.
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Naszódi, M. Approximating a Convex Body by a Polytope Using the Epsilon-Net Theorem. Discrete Comput Geom 61, 686–693 (2019). https://doi.org/10.1007/s00454-018-9977-0
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DOI: https://doi.org/10.1007/s00454-018-9977-0