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

Henze, Matthias; Malikiosis, Romanos Diogenes

doi:10.1007/s00010-016-0458-3

Cited by 9 publications

(22 citation statements)

References 13 publications

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“…, z d ). Then, since by (18) we know that tz / ∈ O for t > 0 large enough, the supremum t z of t > 0 such that tz ∈ O is well-defined and we have (see Figure 11)…”

Section: Checking the Covering Property Over Compact Setsmentioning

confidence: 96%

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On the time for a runner to get lonely

Rifford¹

2021

Preprint

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The Lonely Runner Conjecture asserts that if n runners with distinct constant speeds run on the unit circle R/Z starting from 0 at time 0, then each runner will at some time t > 0 be lonely in the sense that she/he will be separated by a distance at least 1/n from all the others at time t. In investigating the size of t, we show that an upper bound for t in terms of a certain number of rounds (which, in the case where the lonely runner is static, corresponds to the number of rounds of the slowest non-static runner) is equivalent to a covering problem in dimension n − 2. We formulate a conjecture regarding this covering problem and prove it to be true for n = 3, 4, 5, 6. We also show that the so-called gap of loneliness in one round, where we have m + 1 runners including one static runner, is bounded from below by 1/(2m − 1) for all integer m ≥ 2.

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“…, z d ). Then, since by (18) we know that tz / ∈ O for t > 0 large enough, the supremum t z of t > 0 such that tz ∈ O is well-defined and we have (see Figure 11)…”

Section: Checking the Covering Property Over Compact Setsmentioning

confidence: 96%

“…Remark 1. Our approach is reminiscent to those of Chen [8,9,10,11,12] (see for example [9, Lemma 1 p. 182]) and Beck, Hosten and Schymura [3] (see for example Proposition 1 p. 3) who work in dimension d + 1 and with integral speeds (see also [18]).…”

Section: Feathers Bridges Kwais and Beamsmentioning

confidence: 99%

On the time for a runner to get lonely

Rifford¹

2021

Preprint

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“…In the context of the so-called flatness theorem it also proved crucial in Lenstra's landmark paper [19] on solving Linear Integer Programming in fixed dimension in polynomial time (see [17] for more on the flatness theorem). More recent applications of the covering radius include (a) the classification of lattice polytopes in small dimensions (see [15] and references therein), (b) distances between optimal solutions of mixed-integer programs and their linear relaxations [23], (c) unique-lifting properties of maximal lattice-free polyhedra [1], and (d) another viewpoint on the famous Lonely Runner Problem [14].…”

Section: Introductionmentioning

confidence: 99%

The Covering Radius and a Discrete Surface Area for Non-Hollow Simplices

Codenotti

Santos

Schymura

2021

Discrete Comput Geom

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We explore upper bounds on the covering radius of non-hollow lattice polytopes. In particular, we conjecture a general upper bound of d/2 in dimension d, achieved by the “standard terminal simplices” and direct sums of them. We prove this conjecture up to dimension three and show it to be equivalent to the conjecture of González-Merino and Schymura (Discrete Comput. Geom. 58(3), 663–685 (2017)) that the d-th covering minimum of the standard terminal n-simplex equals d/2, for every $$n\ge d$$ n ≥ d . We also show that these two conjectures would follow from a discrete analog for lattice simplices of Hadwiger’s formula bounding the covering radius of a convex body in terms of the ratio of surface area versus volume. To this end, we introduce a new notion of discrete surface area of non-hollow simplices. We prove our discrete analog in dimension two and give strong evidence for its validity in arbitrary dimension.

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“…. , n k , but it can be relaxed to the rational and thus integral case [5,15]. The lower bound 1 k+1 is best possible, as the case n j = j for 0 ≤ j ≤ k and a classic result of Dirichlet on Diophantine approximation (see, e.g., [6]) show.…”

Section: Introductionmentioning

confidence: 99%

“…In the next section, we introduce this model by defining the lonely runner polyhedron P(n). It turns out to be closely related to the zonotopes that where constructed in [15]. We illustrate the utility of a polyhedral ansatz in Section 3 by providing geometric proofs of some folklore results that are only implicit in the existing literature, and by obtaining a new family of lonely runner instances in Theorem 5 defined by the parities of the speeds.…”

Section: Introductionmentioning

confidence: 99%

Lonely Runner Polyhedra

Beck,

Hosten,

Schymura

2016

Preprint

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We study the Lonely Runner Conjecture, conceived by Jörg M. Wills in the 1960's: Given positive integers n 1 , n 2 , . . . , n k , there exists a positive real number t such that for all 1 ≤ j ≤ k the distance of t n j to the nearest integer is at least 1 k+1 . Continuing a view-obstruction approach by Cusick and recent work by Henze and Malikiosis, our goal is to promote a polyhedral ansatz to the Lonely Runner Conjecture. Our results include geometric proofs of some folklore results that are only implicit in the existing literature, a new family of affirmative instances defined by the parities of the speeds, and geometrically motivated conjectures whose resolution would shed further light on the Lonely Runner Conjecture.

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

Cited by 9 publications

References 13 publications

On the time for a runner to get lonely

On the time for a runner to get lonely

The Covering Radius and a Discrete Surface Area for Non-Hollow Simplices

Lonely Runner Polyhedra

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