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arXiv · 1606.01783

Lonely Runner Polyhedra

Matthias Beck·Serkan Hosten·Matthias Schymura
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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.

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Matthias Beck, Serkan Hosten, Matthias Schymura. 2016-06-06. Lonely Runner Polyhedra. https://arxiv.org/abs/1606.01783

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