On Unit Distances in a Convex Polygon

TL;DR

This paper improves the upper bound on the number of unit distances in convex polygons from 2πn log₂ n + O(n) to n log₂ n + O(n), introduces forbidden cycle configurations, and explores vertex arrangements.

Contribution

It provides a tighter upper bound on unit distances in convex polygons and identifies forbidden configurations, advancing understanding of geometric distance problems.

Findings

Improved upper bound to n log₂ n + O(n)

Identified forbidden cycle configurations in convex polygons

Established limitations of current methods through lower bounds

Abstract

In 1959, Erd\H{o}s and Moser asked for the maximum number of unit distances that may be formed among the vertices of a convex $n$-gon; until now, the best known upper bound has been $2\pi n \log_2 n + O(n)$, achieved by F\"uredi in 1990. In this paper, we examine two properties that any convex polygon must satisfy and use them to prove several new facts related to the question posed by Erd\H{o}s and Moser. In particular, we improve on F\"uredi's result, and instead obtain a bound of $n \log_2 n + O(n)$; we exhibit a class of `cycles' formed by unit distances that are forbidden in convex polygons; and we provide a lower bound that shows the limitations of our methods. The second result addresses a question asked by Fishburn and Reeds regarding the possible configurations of vertices that form a convex polygon.