The maximum number of unit distances in a convex n-gon

We show that the maximum number of unit distances or of diameters in a set of n points in d-dimensional Euclidean space is attained only by specific types of Lenz constructions, for all dź4 and n sufficiently large depending on d. As a corollary, we ...

We initiate the study of spanners under the Hausdorff and Fréchet distances. Let S be a set of points in R d and ε a non-negative real number. A subgraph H of the Euclidean graph over S is an ε-Hausdorff-spanner (resp., an ε-Fréchet-spanner) of S,...

• We initiate the study of spanners under the Hausdorff and Fréchet distances.
• We show that any t-spanner of a planar point-set S is a f(t)-Hausdorff-spanner and a g(t)-Fréchet spanner.
• We prove that for any t > 1, there exist a set ...

In 1959, Erd s and Moser asked for the maximum number of unit distances that may occur among the vertices of a convex n -gon. Until now, the best known upper bound has been 2 π n log 2 n + O ( n ) , achieved by Füredi in 1990. In this paper we examine ...