On the covering radius of lattice zonotopes and its relation to view-obstructions and the lonely runner conjecture

TL;DR

This paper explores the connection between geometric problems like view-obstruction and billiard motions with lattice zonotope covering radii, providing bounds via the Flatness Theorem, and relates these to the lonely runner conjecture.

Contribution

It establishes the equivalence between geometric problems and lattice zonotope covering radii estimation, and applies the Flatness Theorem to derive upper bounds.

Findings

Established equivalence between view-obstruction, billiard motions, and lattice zonotope covering radii.

Derived upper bounds for covering radii using the Flatness Theorem.

Linked these geometric problems to the lonely runner conjecture.

Abstract

The goal of this paper is twofold; first, show the equivalence between certain problems in geometry, such as view-obstruction and billiard ball motions, with the estimation of covering radii of lattice zonotopes. Second, we will estimate upper bounds of said radii by virtue of the Flatness Theorem. These problems are similar in nature with the famous lonely runner conjecture.