Shifted Lonely Runner Conjecture (covering radius formulation)

Prove that for every integer n ≥ 1 and every lattice zonotope Z in Z^{n−1} generated by n vectors in general position, the covering radius of Z with respect to Z^{n−1} satisfies µ(Z − x) = (n−1)/(n+1), where x is the center of the zonotope. This is the shifted variant where runners may start at distinct positions.

Background

The shifted version allows distinct starting positions and translates naturally into statements about covering radii of zonotopes. It is known for n = 2 and n = 3 via geometric methods, and reductions suggest finite checking suffices under certain bounds.

The survey emphasizes geometric interpretations that bridge view–obstruction and polytopal formulations, and notes related open problems (e.g., the Lonely Vector Problem) connected to higher-dimensional reductions.

References

The corresponding conjecture, which postulates the same bound in this case is explicitly formulated by Beck, Hosten and Schymura [7]. … Following the zonotopal interpretation of the LR Problem described in Subsection 3.3, the shifted LR Problem can be phrased in terms of covering radii as following: for every lattice zonotope Z generated by n vectors in general position in Zn−1, µ Z−x = n−1 n+1 .

— The Lonely Runner Conjecture turns 60 (2409.20160 - Perarnau et al., 30 Sep 2024) in Section 10.3 (Shifted LR Conjecture)