Coloopless zonotopes and counterexamples to the Shifted Lonely Runner Conjecture

TL;DR

This paper explores the geometric aspects of the Lonely Runner Conjecture and its shifted version, introducing coloopless zonotopes and providing counterexamples that challenge existing assumptions.

Contribution

It introduces coloopless zonotopes, generalizes existing theorems, and presents explicit counterexamples to the shifted LRC and the Lonely Vector Property.

Findings

Counterexamples to the shifted LRC starting at n=5

Counterexamples to the Lonely Vector Property starting at n=12

Generalization of zonotopal approaches to LRC using coloopless zonotopes

Abstract

Henze and Malikiosis (2017) have shown that the Lonely Runner Conjecture (LRC) can be restated as a convex-geometric question on the so-called LR zonotopes, lattice zonotopes with one more generator than their dimension. This relation naturally suggests a more generel statement, the "shifted" LRC, the zonotopal version of which concerns a classical parameter, the covering radius. Theorems A and B in Malikiosis-Schymura-Santos (2025) use the zonotopal restatements of both the original and the shifted LRC to prove a linearly-exponential bound on the size of the (integer) speeds for which the conjectures need to be checked in order to establish them for each fixed number of runners; in the shifted version their statement and proof rely on a certain assumption on two-dimensional rational vector configurations, the so-called "Lonely Vector Property". In this paper we do two things: We push the analogies between the two versions of LRC and their zonotopal counterparts, in particular highlighting that the proofs of Theorems A and B in Malikiosis-Schymura-Santos are more transparent, and the statments more general, if regarded in terms of two quite general classes of lattice zonotopes: the coloopless zonotopes that we introduce here and the cosimple ones, already defined by them. These classes contain all primitive zonotopes of widths at least two and at least three, respectively. We show explicit counterexamples to both the shifted Lonely Runner Conjecture (starting at $n=5$) and to the Lonely Vector Property (starting at $n=12$).