Lonely Runner Polyhedra

2016/06/06 by Matthias Beck, Serkan Hosten, Matthias Schymura · 2 voices · 2 citations
Mathematics · #math.CO #math.NT

paper · pdf

Abstract

We study the Lonely Runner Conjecture, conceived by Jörg M.~Wills in the 1960's: Given positive integers n₁, n₂, …, n_k, there exists a positive real number t such that for all 1 ≤ j ≤ k the distance of t n_j to the nearest integer is at least ( 1 )/( k+1 ). Continuing a view-obstruction approach by Cusick and recent work by Henze and Malikiosis, our goal is to promote a polyhedral 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.

Cited by

• Linear independence of time-frequency translates for ultimately positive functions
• Covering radii of 3-zonotopes and the shifted Lonely Runner Conjecture

Discussions

• Lonely Runner Conjecture may be proved [hn, 5 points, 1 comments]
• Every runner is sometimes lonely [hn, 2 points, 1 comments]

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