The zonotopal shifted Lonely Runner Conjecture

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Let ZvZ_{\mathbf v} be a Strong Lonely Runner Zonotope, meaning a Lonely Runner Zonotope associated with a velocity vector v=(v1,…,vn)∈Z>0n\mathbf v=(v_1,\dotsc,v_n)\in\mathbb{Z}_{>0}^n whose entries are pairwise distinct. Zonotopal shifted Lonely Runner Conjecture. Every Strong Lonely Runner Zonotope of dimension n−1n-1 satisfies

μ(Zv)≤n−1n+1,\mu(Z_{\mathbf v})\leq\frac{n-1}{n+1},

where μ\mu denotes the covering radius with respect to Zn−1\mathbb{Z}^{n-1}. This is the covering-radius reformulation of the shifted Lonely Runner Conjecture and is stronger than the corresponding centered lattice-point assertion. Its general validity remains open.

References

Primary source

Romanos Diogenes Malikiosis, Francisco Santos and Matthias Schymura, “Linearly-exponential checking is enough for the Lonely Runner Conjecture and some of its variants”, arXiv:2411.06903 (2025).

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