Counting large distances in convex polygons: A computational approach

Abstract

In a convex n-gon, let d1 > d2 > ⋯ denote the set of all distances between pairs of vertices, and let mi be the number of pairs of vertices at distance di from one another. Erdös, Lovász, and Vesztergombi conjectured that ∑i≤k mi ≤ kn. Using a new computational approach, we prove their conjecture when k ≤ 4 and n is large; we also make some progress for arbitrary k by proving that ∑i≤k mi ≤ (2k - 1)n. Our main approach revolves around a few known facts about distances, together with a computer program that searches all distance configurations of two disjoint convex hull intervals up to some finite size. We thereby obtain other new bounds, such as m3 ≤ 3n/2 for large n.