Lonely Runner Polyhedra

We study the Lonely Runner Conjecture, conceived by Jörg M. Wills in the 1960's: Given positive integers n_1, n_2, ... , 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 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.


Published in:
Integers
Year:
Sep 04 2018
Keywords:
Dataset(s):
url: https://arxiv.org/abs/1606.01783
Laboratories:




 Record created 2018-09-21, last modified 2019-06-19


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