Counting large distances in convex polygons

TL;DR

This paper proves a conjecture about the sum of the counts of pairs of vertices at large distances in convex polygons for certain cases, using computational methods to establish new bounds and progress on the problem.

Contribution

It introduces a computational approach to prove a longstanding conjecture for small k and large n, and provides new bounds for distances in convex polygons.

Findings

Proved the conjecture for k <= 4 and large n.

Established that m[1]+...+m[k] <= (2k-1)n for arbitrary k.

Derived new bounds such as m[3] <= 3n/2 for large n.

Abstract

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. Erdos, Lovasz, and Vesztergombi conjectured that m[1] + ... + m[k] <= k*n. 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 m[1] + ... + m[k] <= (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 m[3] <= 3n/2 for large n.