On the Fejes Tóth problem about the sum of angles between lines

Bilyk, Dmitriy; Matzke, Ryan

doi:10.1090/proc/14263

Cited by 15 publications

(13 citation statements)

References 12 publications

(18 reference statements)

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“…where to obtain (11) we used (9), and where u on the last line is any fixed point in X . Substituting (9) and (11) into (8) and rearranging, we obtain (7).…”

mentioning

confidence: 99%

Stolarsky's Invariance Principle for Finite Metric Spaces

Barg

2020

Mathematika

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Stolarsky's invariance principle quantifies the deviation of a subset of a metric space from the uniform distribution. Classically derived for spherical sets, it has been recently studied in a number of other situations, revealing a general structure behind various forms of the main identity. In this work we consider the case of finite metric spaces, relating the quadratic discrepancy of a subset to a certain function of the distribution of distances in it. Our main results are related to a concrete form of the invariance principle for the Hamming space. We derive several equivalent versions of the expression for the discrepancy of a code, including expansions of the discrepancy and associated kernels in the Krawtchouk basis. Codes that have the smallest possible quadratic discrepancy among all subsets of the same cardinality can be naturally viewed as energy-minimizing subsets in the space. Using linear programming, we find several bounds on the minimal discrepancy and give examples of minimizing configurations. In particular, we show that all binary perfect codes have the smallest possible discrepancy.

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“…where to obtain (11) we used (9), and where u on the last line is any fixed point in X . Substituting (9) and (11) into (8) and rearranging, we obtain (7).…”

mentioning

confidence: 99%

Stolarsky's Invariance Principle for Finite Metric Spaces

Barg

2020

Mathematika

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“…Not only is Proposition 1.1 limited to even p's, but it is also not trivial to find spherical t-designs for large t. More generally, and to the best of our knowledge, little is known about the complete solutions to (4) even in the simplest case d = 2. When N = 3, a solution is given in [13] for all positive p. See also [6,19] for related results. For any N and p = ∞, it is shown in [5] that the Grassmannian frame is (10) X…”

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confidence: 99%

Universal optimal configurations for the $p$-frame potentials

Gonzales¹,

Goodman²,

Kang³

et al. 2019

Preprint

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Given d, N ≥ 2 and p ∈ (0, ∞] we consider a family of functionals, the p-frame potentials FP p,N,d , defined on the set of all collections of N unit-norm vectors in R d . For the special case p = 2 and p = ∞, both the minima and the minimizers of these potentials have been thoroughly investigated. In this paper, we investigate the minimizers of the functionals FP p,N,d , by first establishing some general properties of their minima. Thereafter, we focus on the special case d = 2, for which, surprisingly, not much is known. One of our main results establishes the unique minimizer for big enough p. Moreover, this minimizer is universal in the sense that it minimizes a large range of energy functions that includes the p-frame potential. We conclude the paper by reporting some numerical experiments for the case d ≥ 3, N = d + 1, p ∈ (0, 2). These experiments lead to some conjectures that we pose.

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“…He proved this conjecture for at most 6 lines, and he also gave an upper bound for the sum of angles for an arbitrary number of lines. Our main result of Chapter 2 is Theorem 2.1 in which we give an upper bound for the sum of the angles of n lines in 3-dimensional space, thus improving the general bound of L. Fejes Tóth for large n. We note that after our paper had been published, Bilyk and Matzke [BM19] further improved the upper bound using a different method.…”

Section: Chapter 1 Forewordmentioning

confidence: 79%

“…This means that the sum of angles is asymptotically less than n 2 π/5 as n → ∞. In our paper [FVZ16b], we improved this upper bound to 3n 2 π/16 ≈ 0.589 • n 2 , and later Bilyk and Matzke [BM19] further improved it to π 4 − 69 100d n 2 as n → ∞. We note that their result for d = 3 gives asymptotically less than 0.556 • n 2 as n → ∞.…”

Section: Proof Of Theorem 57mentioning

confidence: 85%

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Convex and Discrete Geometrical Problems on the Sphere

Zarnócz

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On the Fejes Tóth problem about the sum of angles between lines

Cited by 15 publications

References 12 publications

Stolarsky's Invariance Principle for Finite Metric Spaces

Stolarsky's Invariance Principle for Finite Metric Spaces

Universal optimal configurations for the $p$-frame potentials

Convex and Discrete Geometrical Problems on the Sphere

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