On the diameter of separated point sets with many nearly equal distances
نویسندگان
چکیده
منابع مشابه
On the diameter of separated point sets with many nearly equal distances
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. ...
متن کاملOn Point Sets with Many Unit Distances in Few Directions
We study the problem of the maximum number of unit distances among n points in the plane under the additional restriction that we count only those unit distances that occur in a xed set of k directions taking the maximum over all sets of n points and all sets of k directions We prove that for xed k and su ciently large n n k the extremal sets are essentially sections of lattices bounded by edge...
متن کاملNearly 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...
متن کاملNearly equal distances in metric spaces
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...
متن کاملOn the minimum diameter of plane integral point sets
Since ancient times mathematicians consider geometrical objects with integral side lengths. We consider plane integral point sets P , which are sets of n points in the plane with pairwise integral distances where not all the points are collinear. The largest occurring distance is called its diameter. Naturally the question about the minimum possible diameter d(2, n) of a plane integral point se...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: European Journal of Combinatorics
سال: 2006
ISSN: 0195-6698
DOI: 10.1016/j.ejc.2006.05.007