Lonely Runner Polyhedra

TL;DR

This paper explores the Lonely Runner Conjecture using a geometric polyhedral approach, providing new proofs, affirmative cases based on speed parities, and conjectures to advance understanding of the problem.

Contribution

It introduces a polyhedral framework for the conjecture, proves some folklore results geometrically, and identifies new affirmative instances and conjectures.

Findings

Geometric proofs of folklore results

New affirmative cases based on speed parities

Conjectures that could further illuminate the conjecture

Abstract

We study the \emph{Lonely Runner Conjecture}, conceived by J\"org M.~Wills in the 1960's: Given positive integers $n_1, n_2, \dots, n_k$, there exists a positive real number $t$ such that for all $1 \le j \le k$ the distance of $t \, n_j$ to the nearest integer is at least $\frac{ 1 }{ k+1 }$. Continuing a view-obstruction approach by Cusick and recent work by Henze and Malikiosis, our goal is to promote a polyhedral \emph{ansatz} to the Lonely Runner Conjecture. Our results include geometric proofs of some folklore results that are only implicit in the existing literature, a new family of affirmative instances defined by the parities of the speeds, and geometrically motivated conjectures whose resolution would shed further light on the Lonely Runner Conjecture.