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**Zentralblatt MATH**

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

**Zbl.No: ** 722.52009

**Autor: ** Erdös, Paul; Pach, János

**Title: ** Variations on the theme of repeated distances. (In English)

**Source: ** Combinatorica 10, No.3, 261-269 (1990).

**Review: ** Given a set X = **{**x_{1},...x_{n}**}** of n points in **R**^{d}, let f(X) denote the number of pairs **{**x_{i},x_{j}**}** whose Euclidean distances ||x_{i}-x_{j}|| = 1. Let f_{d}(n) = **max**_{X\subset Rd, |X| \leq n}f(X). An asymptotically sharp estimate for the error term in this maximum is given for d \geq 4. Tight upper bounds are also determined for the total number of occurrences of what are called ``favourite'' distances from n points in **R**^{d}, d \geq 4. Some related results are also proved for distances determined by n disjoint compact convex sets in **R**^{2}.

**Reviewer: ** P.Smith (Keele)

**Classif.: ** * 52C10 Erdoes problems and related topics of discrete geometry

52A10 Convex sets in 2 dimensions (including convex curves)

52A20 Convex sets in n dimensions

**Keywords: ** repeated distances; points in **R**^{d}; convex sets

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag