The zonotopal Lonely Runner Conjecture

Let $Z=\sum_{i=1}^n[\mathbf{0},\mathbf{u}_i]\subseteq\mathbb{R}^{n-1}$ be a lattice zonotope of dimension $n-1$ with $n$ generators, whose every $n-1$ generators are linearly independent; such a zonotope is a Lonely Runner Zonotope. Let

$$\mathbf{x}=\frac{1}{2}(\mathbf{u}_1+\dotsb+\mathbf{u}_n)$$

be its center. Zonotopal Lonely Runner Conjecture. For every Lonely Runner Zonotope $Z$,

$$\frac{n-1}{n+1}(Z-\mathbf{x})\cap(\mathbf{x}+\mathbb{Z}^{n-1})\neq\varnothing.$$

This is a geometric reformulation of the Lonely Runner Conjecture. The paper uses it to translate the problem into lattice-point and covering questions for zonotopes; its general validity remains open.