### Abstract

Let s_{1} (n) denote the largest possible minimal distance among n distinct points on the unit sphere {Mathematical expression}. In general, let s_{k}(n) denote the supremum of the k-th minimal distance. In this paper we prove and disprove the following conjecture of A. Bezdek and K. Bezdek: s_{2}(n) = s_{1}([n/3]). This equality holds for n > n_{0} however s_{2}(12) > s_{1}(4). We set up a conjecture for s_{k}(n), that one can always reduce the problem of the k-th minimum distance to the function s_{1}. We prove this conjecture in the case k=3 as well, obtaining that s_{3}(n) = s_{1}([n/5]) for sufficiently large n. The optimal construction for the largest second distance is obtained from a point set of size [n/3] with the largest possible minimal distance by replacing each point by three vertices of an equilateral triangle of the same size e{open}. If e{open} → 0, then s_{2} tends to s_{1}([n/3]). In the case of the third minimal distance, we start with a point set of size [n/5] and replace each point by a regular pentagon.

Original language | English |
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Pages (from-to) | 55-65 |

Number of pages | 11 |

Journal | Journal of Geometry |

Volume | 46 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - Mar 1993 |

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### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

**The second and the third smallest distances on the sphere.** / Füredi, Z.

Research output: Contribution to journal › Article

*Journal of Geometry*, vol. 46, no. 1-2, pp. 55-65. https://doi.org/10.1007/BF01231000

}

TY - JOUR

T1 - The second and the third smallest distances on the sphere

AU - Füredi, Z.

PY - 1993/3

Y1 - 1993/3

N2 - Let s1 (n) denote the largest possible minimal distance among n distinct points on the unit sphere {Mathematical expression}. In general, let sk(n) denote the supremum of the k-th minimal distance. In this paper we prove and disprove the following conjecture of A. Bezdek and K. Bezdek: s2(n) = s1([n/3]). This equality holds for n > n0 however s2(12) > s1(4). We set up a conjecture for sk(n), that one can always reduce the problem of the k-th minimum distance to the function s1. We prove this conjecture in the case k=3 as well, obtaining that s3(n) = s1([n/5]) for sufficiently large n. The optimal construction for the largest second distance is obtained from a point set of size [n/3] with the largest possible minimal distance by replacing each point by three vertices of an equilateral triangle of the same size e{open}. If e{open} → 0, then s2 tends to s1([n/3]). In the case of the third minimal distance, we start with a point set of size [n/5] and replace each point by a regular pentagon.

AB - Let s1 (n) denote the largest possible minimal distance among n distinct points on the unit sphere {Mathematical expression}. In general, let sk(n) denote the supremum of the k-th minimal distance. In this paper we prove and disprove the following conjecture of A. Bezdek and K. Bezdek: s2(n) = s1([n/3]). This equality holds for n > n0 however s2(12) > s1(4). We set up a conjecture for sk(n), that one can always reduce the problem of the k-th minimum distance to the function s1. We prove this conjecture in the case k=3 as well, obtaining that s3(n) = s1([n/5]) for sufficiently large n. The optimal construction for the largest second distance is obtained from a point set of size [n/3] with the largest possible minimal distance by replacing each point by three vertices of an equilateral triangle of the same size e{open}. If e{open} → 0, then s2 tends to s1([n/3]). In the case of the third minimal distance, we start with a point set of size [n/5] and replace each point by a regular pentagon.

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UR - http://www.scopus.com/inward/citedby.url?scp=34250083646&partnerID=8YFLogxK

U2 - 10.1007/BF01231000

DO - 10.1007/BF01231000

M3 - Article

VL - 46

SP - 55

EP - 65

JO - Journal of Geometry

JF - Journal of Geometry

SN - 0047-2468

IS - 1-2

ER -