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 Z1, Z2, ... be any sequence of points on the plane. Then there is a point X in E for which infinitely many of the products \overline{XZ1}· ... · \overline{XZn} 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

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