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Coloopless and cosimple zonotopes, and the Lonely Runner Conjectures

Mónica Blanco, Francisco Criado, Francisco Santos

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 \emph{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: 1. 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. 2. We show explicit counterexamples to both the shifted Lonely Runner Conjecture (starting at $n=5$) and to the Lonely Vector Property (starting at $n=12$).

Coloopless and cosimple zonotopes, and the Lonely Runner Conjectures

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 \emph{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: 1. 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. 2. We show explicit counterexamples to both the shifted Lonely Runner Conjecture (starting at ) and to the Lonely Vector Property (starting at ).

Paper Structure

This paper contains 17 sections, 56 theorems, 66 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Covering and central radii of convex bodies
  4. Relation to successive minima
  5. Zonotopal restatements of LRC
  6. Coloopless and cosimple zonotopes
  7. LRC and coloopless zonotopes. Revisiting the volume upper bound
  8. Every coloopless zonotope contains an LR-zonotope.
  9. Some non-coloopless (counter)-examples
  10. Shifted LRC, cosimple zonotopes, and the Lonely Vector Property
  11. Minimal cosimple polytopes and the Lonely Vector Property
  12. Vector configurations without the Lonely Vector Property
  13. All cosimple zonotopes properly contain zonotopes of width at least three
  14. Counterexamples to the shifted LRC
  15. The algorithm
  16. ...and 2 more sections

Key Result

Lemma 1.4

Let $\mathbf{v}$ be as in Definition def:loneliness. Let $\mathbf{\tfrac{1}{2}}:=(\tfrac{1}{2},\dots,\tfrac{1}{2})\in \mathbb{R}^n$, and let $\mathop{\mathrm{dist}}\nolimits_\infty(\cdot, \cdot)$ denote the $L_\infty$ distance in $\mathbb{R}^n$. Then:

Figures (6)

  • Figure 1: Distance to the origin of five runners with $\mathbf{v}=(1,2,3,4,5)$ and $\mathbf{s}=\tfrac{1}{94}(0,46,38,47,72)$. The dashed line is $\{y=\tfrac{15}{94}\}$, and the fact that at every moment in time there is some runner on or below that line shows that $\gamma^{\min}(1,2,3,4,5) \leq \tfrac{15}{94} < \tfrac{1}{6}$. The four dots along this line are the instants when the minimum distance from the runners to the origin equals$\tfrac{15}{94}$, which implies this minimum to be attained by two (or more) runners.
  • Figure 2: The parameters $\kappa$ and $\mu$ of the zonotopes $P$ in Table \ref{['table:kappas']}
  • Figure 3: The parameters $\lambda_1$ and $\lambda_2$ of the zonotopes $P-P$ in Table \ref{['table:kappas']}. Observe that changing from $P$ to $P-P$ for a centrally symmetric $P$ is equivalent to refining the lattice by a factor of two and centering $P$ at the origin.
  • Figure 4: Illustration of the proof of Lemma \ref{['lemma:contained-diagonal']}
  • Figure 5: Left: the symmetrized configuration $\overline S$ of a counterexample $S$ to LVP with $|S|=12$. Right: same with $|S|=16$
  • ...and 1 more figures

Theorems & Definitions (107)

  • Remark 1.2
  • Definition 1.3
  • Lemma 1.4
  • proof
  • Corollary 1.5: sixrunnerszonorunners
  • Corollary 1.6
  • Lemma 1.7
  • Proposition 1.8
  • Remark 1.10
  • Theorem 1.11: Malikiosis2024LinExpCheckingLRC
  • ...and 97 more