Nearly Equal Distances in the Plane
نویسندگان
چکیده
منابع مشابه
Nearly Equal Distances in the Plane
distances determined by them? In particular, what is the maximum number of pairs of points that determine the same distance? Although a lot of progress has been made in this area, we are still very far from having satisfactory answers to the above questions (cf. [EP], [MP], [PA] for recent surveys). Two distances are said to be nearly the same if they differ by at most 1. If all points of a set...
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Let (X, d) be any finite metric space with n elements. We show that there are two pairs of distinct elements in X that determine two nearly equal distances in the sense that their ratio differs from 1 by at most 9 logn n2 . This bound (apart for the multiplicative constant) is best possible and we construct a metric space that attains this bound. We discus related questions and consider in part...
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A point set is separated if the minimum distance between its elements is 1. We call two real numbers nearly equal if they differ by at most 1. We prove that for any dimension d ≥ 2 and any γ > 0, if P is a separated set of n points in Rd such that at least γ n2 pairs in ( P 2 ) determine nearly equal distances, then the diameter of P is at least C(d, γ )n2/(d−1) for some constant C(d, γ ) > 0. ...
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ژورنال
عنوان ژورنال: Combinatorics, Probability and Computing
سال: 1993
ISSN: 0963-5483,1469-2163
DOI: 10.1017/s0963548300000791