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**Zentralblatt MATH**

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

**Zbl.No: ** 444.52008

**Autor: ** Erdös, Paul; Pach, János

**Title: ** On a problem of L. Fejes Toth. (In English)

**Source: ** Discrete Math. 30, 103-109 (1980).

**Review: ** Let 0 \leq x_{1} \leq x_{2} \leq x_{3} \leq ... be a sequence of real numbers, **lim** x_{i} = +oo. The authors prove that if **sum**_{i}l/x_{i}^{n-k} = +oo then there exists a point-system P = **{**z_{1},z_{2},...**}** in the n-dimensional space **E**^{n}, for which |z_{i}| = x_{i} holds (i = 1,2,...), and any k-dimensional plane comes arbitrarily near to P. this result is best possible in the sense that if P**{**z_{1},z_{2},...**}** is a point-system satisfying **sum**_{i}l/|z_{i}|^{n-k} <+oo then for every C > 0 there exists a k-dimensional plane in BbbE^{n}, whose distance from all members of P is at least C. A generalization is also proved. This settles a problem of *L. Fejes Tóth* [Mat. Lapok 25 (1974), 13-20 (1976; Zbl 359.52010)].

**Classif.: ** * 52A37 Other problems of combinatorial convexity

52A40 Geometric inequalities, etc. (convex geometry)

**Keywords: ** countable point-system in E^{2}; plane comes arbitrarily near to P

**Citations: ** Zbl.359.52010

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