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 = {x1,...xn} of n points in Rd, let f(X) denote the number of pairs {xi,xj} whose Euclidean distances ||xi-xj|| = 1. Let

fd(n) = maxX\subset Rd,  |X| \leq nf(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 Rd, d \geq 4. Some related results are also proved for distances determined by n disjoint compact convex sets in R2.
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 Rd; convex sets

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