Deep lattice points in zonotopes, lonely runners, and lonely rabbits

Abstract

Let $K \subseteq \mathbb{R}^d$ be a convex body and let $\mathbf{w} \in \operatorname{int}(K)$ be an interior point of $K$. The coefficient of asymmetry $\operatorname{ca}(K,\mathbf{w}) := \min\{ \lambda \geq 1 : \mathbf{w} - K \subseteq \lambda (K - \mathbf{w}) \}$ has been studied extensively in the realm of Hensley's conjecture on the maximal volume of a $d$-dimensional lattice polytope that contains a fixed positive number of interior lattice points. We study the coefficient of asymmetry for lattice zonotopes, i.e., Minkowski sums of line segments with integer endpoints. Our main result gives the existence of an interior lattice point whose coefficient of asymmetry is bounded above by an explicit constant in $\Theta(d \log\log d)$, for any lattice zonotope that has an interior lattice point. Our work is both inspired by and feeds on Wills' lonely runner conjecture from Diophantine approximation: we make intensive use of a discrete version of this conjecture, and reciprocally, we reformulate the lonely runner conjecture in terms of the coefficient of asymmetry of a zonotope.