## Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  853.52017
Autor:  Beck, István; Bejlegaard, Niels; Erdös, Paul; Fishburn, Peter
Title:  Equal distance sums in the plane. (In English)
Source:  Normat 43, No.4, 150-161 (1995).
Review:  Let Xn = {x1,...,xn} be a set of n distinct points in the plane; and let Si be the sum of (Euclidean) distances from the point xi to the other points in Xn. Xn is called an equisum set if S1 = ... = Sn.
It is obvious that X3 is an equisum set iff X3 is the vertex set of a regular triangle. In this paper is simply shown that X4 is an equisum set iff its points are the corners of a rectangle. The authors investigate all about the five-point set X5. Here the quasi-convexity and the bilaterally symmetry play a role.
Xn is called quasi-convex if its points are the vertices of a strictly convex n-gon, and bilaterally-symmetric if Xn is identical to the set obtained by rotating its points 180\circ around some line.
All equisum sets Xn are quasi-convex.
Some interesting results:
-- If X5 is a bilaterally symmetric equisum set, then it has exactly 2, 5 or 6 different interpoint distances.
-- If X5 is an equisum set, but not bilaterally symmetric, then it has exactly 8, 9, or 10 different interpoint distances.
In the cases that the number of interpoint distances is 6, 9 or 10 there are infinitely many dissimilar equisum sets X5.
This paper is related to papers by P. Erdös and P. Fishburn [Discrete Appl. Math. 60, No. 1-3, 149-158 (1995; Zbl 831.52009)], by V. Klee and St. Wagon [`Old and new unsolved problems in plane geometry and number theory' (1991; Zbl 784.51002)] and others.
Reviewer:  E.Quaisser (Potsdam)
Classif.:  * 52C10 Erdoes problems and related topics of discrete geometry
52A40 Geometric inequalities, etc. (convex geometry)
51M16 Inequalities and extremum problems (geometry)
Keywords:  Erdös problem; minimal number of points; equisum set; distinct distances in finite point sets
Citations:  Zbl 831.52009; Zbl 784.51002

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