### Abstract

Let x_{1},...,x_{n} be n distinct points in the plane. Denote by D(x_{1},...,x_{n}) the minimum number of distinct distances determined by x_{1},...,x_{n}. Put f{hook}(n) = min D(x_{1},..., x_{n}). An old and probably very difficult conjecture of mine states that f{hook}(n) > cn (log n)^{ 1 2}. f{hook}(5) = 2 and the only way we can get f{hook}(5) = 2 is if the points form a regular pentagon. Are there other values of n for which there is a unique configuration of points for which the minimal value of f{hook}(n) is assumed? Is it true that the set of points which implements f{hook}(n) has lattice structure? Many related questions are discussed.

Original language | English |
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Pages (from-to) | 147-153 |

Number of pages | 7 |

Journal | Discrete Mathematics |

Volume | 60 |

Issue number | C |

DOIs | |

Publication status | Published - Jan 1 1986 |

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### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics