Counting large distances in convex polygons: a computational approach

In a convex n-gon, let d 1 > d 2 > ⋯ denote the set of all distances between pairs of vertices, and let m i be the number of pairs of vertices at distance d i from one another. We prove that ∑ i ⩽ k m i ⩽ ( 2 k − 1 ) n , making progress towards the conjecture ∑ i ⩽ k m i ⩽ k n of Erdős, Lovász, and Vesztergombi. Using a new computational approach, for large enough n, we prove that ∑ i ⩽ 3 m i ⩽ 3 n and ∑ i ⩽ 4 m i ⩽ 4 n , partially resolving their conjecture, as well as m 3 ⩽ 3 2 n and related bounds. Our abstraction uses a computer program to search all small configurations of distances generated by two disjoint intervals.