Counting large distances in convex polygons

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.

💡 Research Summary

The paper studies a classic extremal problem in discrete geometry: given a set S of n points that form the vertices of a convex polygon, list all pairwise distances and order them decreasingly as d₁ > d₂ > … . Let m_i denote the number of unordered vertex pairs whose distance equals d_i. A conjecture of Erdős, Lovász and Vesztergombi (1990s) asserts that for any integer k ≥ 1 the total number of occurrences of the k largest distances satisfies

  m₁ + m₂ + … + m_k ≤ k·n.

The conjecture is known to be tight for regular odd‑gon constructions, and it has been proved only for k = 1 (the diameter) and k = 2 (the second‑largest distance) in the convex setting. The present work makes two major advances.

First, the authors introduce a “level” decomposition of the complete graph on the polygon’s vertices. Number the vertices clockwise as a₁,…,a_n; a diagonal a_j a_k belongs to level L_i where i ≡ j + k (mod n). Each level consists of diagonals that are parallel in the auxiliary regular n‑gon. Using a simple geometric inequality (Fact 3.1) together with a known bound on the number of long diagonals in a convex quadrilateral (Fact 3.2), they prove Lemma 3.5: any level contains at most 2k − 1 diagonals of length at least d_k. Since there are exactly n levels, this yields the universal inequality

  m₁ + … + m_k ≤ (2k − 1)·n  (Theorem 1.3).

Thus the conjectured bound is achieved up to a factor of roughly two.

Second, the paper presents a computer‑assisted exhaustive search that handles the remaining combinatorial complexity for small k. The search works on an abstract configuration consisting of two disjoint intervals of vertices (called “top” and “bottom”). For each pair (t_i, b_j) the algorithm maintains a set D