On some metric and combinatorial geometric problems

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Abstract

Let x1,...,xn be n distinct points in the plane. Denote by D(x1,...,xn) the minimum number of distinct distances determined by x1,...,xn. Put f{hook}(n) = min D(x1,..., xn). 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 languageEnglish
Pages (from-to)147-153
Number of pages7
JournalDiscrete Mathematics
Volume60
Issue numberC
DOIs
Publication statusPublished - Jan 1 1986

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ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics

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