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

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• 作者：Konstantin E. Tikhomirov (1)
  
  1. Department of Mathematical and Statistical Sciences ; University of Alberta ; 632 Central Academic Building ; Edmonton ; AB ; T6G2G1 ; Canada
  
• 关键词：Convex polytope ; Banach鈥揗azur distance ; Covering by spherical balls ; 52B11 ; 52A20
• 刊名：Discrete and Computational Geometry
• 出版年：2015
• 出版时间：January 2015
• 年：2015
• 卷：53
• 期：1
• 页码：173-181
• 全文大小：163 KB
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• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Combinatorics
  Computational Mathematics and Numerical Analysis
  
• 出版者：Springer New York
• ISSN：1432-0444

文摘

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 \(c, C\) .