Two and a half billion years of distance research

In a seminal paper published in 1946, Erd ̋os initiated the investigation of the distribution of distances generated by point sets in metric spaces. In spite of some spectacular par- tial successes and persistent attacks by generations of mathe- maticians, most problems raised in Erd ̋os’ paper are still un- solved. GivenasetofnpointsinRd,letd1 >d2 >d3 >··· denote the sequence of all distances between pairs of points in P, listed in decreasing order. We raise some simple ques- tions 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 R2, then the maximum number of equilateral triangles of side length dk induced by P is Θ(k).