The multiplicity of the two smallest distances among points

✍ Scribed by György Csizmadia

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 864 KB
• Volume: 194
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/s0012-365x(98)00116-2

✦ Synopsis

Let 1 = d~ <d2 < ... <dk denote the distinct distances determined by a set of n points in the plane. The multiplicity of the two smallest distances is smaller than 6n and it is maximized by the triangular lattice, where d2 --x/~. We partially answer a question of Erd6s and Vesztergombi by proving that d2 ¢ x/~ implies that the multiplicity of the two smallest distances is at most 4n unless d2 is (v~+ 1)/2 or 1/(2 sinl5). In the case d2 =(v/5+ 1)/2, the multiplicity is at most 4.5n. We also show some extremal configurations for different values of d2.