### Abstract

Let n, N be natural numbers satisfying (Formula presented.) be the unit Euclidean ball in R^{n}, and let (Formula presented.) be a convex n-dimensional polytope with N vertices and the origin in its interior. We prove that (Formula presented.) 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 ϕ, we get that (Formula presented.) for an absolute constant C>0. Both estimates are optimal up to the constant multiples c, C.

Original language | English (US) |
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Pages (from-to) | 173-181 |

Number of pages | 9 |

Journal | Discrete and Computational Geometry |

Volume | 53 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2015 |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

### Keywords

- Banach–Mazur distance
- Convex polytope
- Covering by spherical balls

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## Cite this

Tikhomirov, K. E. (2015). On the Distance of Polytopes with Few Vertices to the Euclidean Ball.

*Discrete and Computational Geometry*,*53*(1), 173-181. https://doi.org/10.1007/s00454-014-9639-9