Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  849.52014
Autor:  Erdös, Paul; Fishburn, Peter
Title:  Convex nonagons with five intervertex distances. (In English)
Source:  Geom. Dedicata 60, No.3, 317-332 (1996).
Review:  The authors prove that the vertices of a convex 9-gon in the euclidean plane determine exactly five distinct distances if and only if they are a subset of the vertices of a regular 10-gon or a regular 11-gon, and conjecture that the vertices of a convex 2m+1-gon determine exactly m+1 distance (m \geq 3) if and only if they are a subset of the vertices of a regular 2m+2- or 2m+3-gon. This may be seen as one step below the extremal situation of Altman's theorem which states that the vertices of a convex n-gon determine at least \lfloor 1/n n \rfloor distinct distances, with equality if and only if the n-gon is regular, or n is even and it is a regular n+1-gon minus one point.
The main reason for the study of point sets with few distinct distances is the famous Erdös conjecture on the minimal number of distinct distances determined by n points in the plane, which is believed to be \Theta ({n \over \sqrt {log n}}) as can be reached by sections of the triangular lattice.
Reviewer:  P.Braß (Greifswald)
Classif.:  * 52C10 Erdoes problems and related topics of discrete geometry
                   52C05 Lattices and convex bodies in 2 dimensions
Keywords:  intervertex distances; convex polygons; Erdös conjecture

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