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On the Distance of Polytopes with Few Vertices to the Euclidean Ball

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Abstract

Let \(n,N\) be natural numbers satisfying \(n+1\le N\le 2n, B_2^n\) be the unit Euclidean ball in \({\mathbb R}^n\), and let \(P\subset B_2^n\) be a convex \(n\)-dimensional polytope with \(N\) vertices and the origin in its interior. We prove that

$$\begin{aligned} \inf \{\lambda \ge 1:\,B_2^n\subset \lambda P\}\ge cn/\sqrt{N-n}, \end{aligned}$$

where \(c>0\) is a universal constant. As an immediate corollary, for any covering of \(S^{n-1}\) by \(N\) spherical caps of geodesic radius \(\phi \), we get that \(\cos \phi \le C\sqrt{N-n}/n\) for an absolute constant \(C>0\). Both estimates are optimal up to the constant multiples \(c, C\).

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Acknowledgments

I would like to thank A.E. Litvak for introducing me to the problem, my supervisor N. Tomczak-Jaegermann for suggestions on the text and U. Yahorau for a useful discussion. Also, I am grateful to the referees for valuable comments and suggestions.

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Correspondence to Konstantin E. Tikhomirov.

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Tikhomirov, K.E. On the Distance of Polytopes with Few Vertices to the Euclidean Ball. Discrete Comput Geom 53, 173–181 (2015). https://doi.org/10.1007/s00454-014-9639-9

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  • DOI: https://doi.org/10.1007/s00454-014-9639-9

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