Two and a half billion years of distance research
Filip Morić, János Pach · 2013
In a seminal paper published in 1946, Erdős initiated the investigation of the distribution of distances generated by point sets in metric spaces. In spite of some spectacular partial successes and persistent attacks by generations of mathematicians, most problems raised in Erdős ’ paper are still unsolved. Given a set of n points in R d, let d1> d2> d3> · · · denote the sequence of all distances between pairs of points in P, listed in decreasing order. We raise some simple questions related to a famous conjecture of Schur. For instance, is it true that any two regular (d − 1)-dimensional simplices of side length d1 induced by P share at least one vertex? We prove that if P is the vertex set of a convex polygon in R 2, then the maximum number of equilateral triangles of side length dk induced by P is Θ(k). 1