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**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 X_{n} = **{**x_{1},...,x_{n}**}** be a set of n distinct points in the plane; and let S_{i} be the sum of (Euclidean) distances from the point x_{i} to the other points in X_{n}. X_{n} is called an equisum set if S_{1} = ... = S_{n}.

It is obvious that X_{3} is an equisum set iff X_{3} is the vertex set of a regular triangle. In this paper is simply shown that X_{4} is an equisum set iff its points are the corners of a rectangle. The authors investigate all about the five-point set X_{5}. Here the quasi-convexity and the bilaterally symmetry play a role.

X_{n} is called quasi-convex if its points are the vertices of a strictly convex n-gon, and bilaterally-symmetric if X_{n} is identical to the set obtained by rotating its points 180^{\circ} around some line.

All equisum sets X_{n} are quasi-convex.

Some interesting results:

-- If X_{5} is a bilaterally symmetric equisum set, then it has exactly 2, 5 or 6 different interpoint distances.

-- If X_{5} 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 X_{5}.

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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