Explicit counterexamples disprove the shifted Lonely Runner Conjecture for n=5 and the Lonely Vector Property for n=12 by introducing coloopless zonotopes.
Covering radii of $3$-zonotopes and the shifted Lonely Runner Conjecture
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abstract
We show that the shifted Lonely Runner Conjecture (sLRC) holds for 5 runners. We also determine that there are exactly 3 primitive tight instances of the conjecture, only two of which are tight for the non-shifted conjecture (LRC). Our proof is computational, relying on a rephrasing of the sLRC in terms of covering radii of certain zonotopes (Henze and Malikiosis, 2017), and on an upper bound for the (integer) velocities to be checked (Malikiosis, Santos and Schymura, 2024+). As a tool for the proof, we devise an algorithm for bounding the covering radius of rational lattice polytopes, based on constructing dyadic fundamental domains.
fields
math.CO 2years
2026 2representative citing papers
Two new upper bounds on covering minima of convex bodies are given from projections and intersections, shown sharp for direct sums and applied to terminal simplices to narrow a 2017 conjecture gap.
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Coloopless zonotopes and counterexamples to the Shifted Lonely Runner Conjecture
Explicit counterexamples disprove the shifted Lonely Runner Conjecture for n=5 and the Lonely Vector Property for n=12 by introducing coloopless zonotopes.
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Upper Bounds on Covering Minima of Convex Bodies
Two new upper bounds on covering minima of convex bodies are given from projections and intersections, shown sharp for direct sums and applied to terminal simplices to narrow a 2017 conjecture gap.