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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 865.41008

**Autor: ** Erdös, Paul; Totik, Vilmos

**Title: ** On the size of products of distances from prescribed points. (In English)

**Source: ** Math. Proc. Camb. Philos. Soc. 120, No.3, 403-409 (1996).

**Review: ** The following problem, raised in a mathematical contest, is investigated:

``Let E be any connected set in the plane of diameter greater than 4, and let Z_{1}, Z_{2}, ... be any sequence of points on the plane. Then there is a point X in E for which infinitely many of the products \overline{XZ_{1}}· ... · \overline{XZ_{n}} are greater than 1. Furthermore, the same is not necessarily true if the diameter of E is 4.''

The problem can be simplified to segments E of length greater than 4. In the paper, the limit case of segments of length 4 is considered. A more precise formulation of the above result for segments greater than 4 is obtained, and the case of more general sets E is studied.

**Reviewer: ** G.Plonka (Rostock)

**Classif.: ** * 41A10 Approximation by polynomials

12E10 Special polynomials over general fields

**Keywords: ** Chebyshev polynomials; Fekete set

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag