On the sum of powered distances to certain sets of points on the circle

Nikolov, Nikolai; Rafailov, Rafael

doi:10.2140/pjm.2011.253.157

Cited by 21 publications

(26 citation statements)

References 3 publications

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“…The following lemma gives two important cases in which the hypothesis (2) from the above proposition holds, the second of which is due to Nikolov and Rafailov [28,Thm. 1.2 (1)].…”

Section: Results Formentioning

confidence: 95%

Unconstrained polarization (Chebyshev) problems: basic properties and Riesz kernel asymptotics

Hardin¹,

Petrache²,

Saff³

2019

Preprint

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We introduce and study the unconstrained polarization (or Chebyshev) problem which requires to find an N -point configuration that maximizes the minimum value of its potential over a set A in p -dimensional Euclidean space. This problem is compared to the constrained problem in which the points are required to belong to the set A . We find that for Riesz kernels 1{|x ´y| s with s ą p ´2 the optimum unconstrained configurations concentrate close to the set A and based on this fundamental fact we recover the same asymptotic value of the polarization as for the more classical constrained problem on a class of d -rectifiable sets. We also investigate the new unconstrained problem in special cases such as for spheres and balls. In the last section we formulate some natural open problems and conjectures.

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“…The following lemma gives two important cases in which the hypothesis (2) from the above proposition holds, the second of which is due to Nikolov and Rafailov [28,Thm. 1.2 (1)].…”

Section: Results Formentioning

confidence: 95%

Unconstrained polarization (Chebyshev) problems: basic properties and Riesz kernel asymptotics

Hardin¹,

Petrache²,

Saff³

2019

Preprint

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“…He also determined M p n for n = 3 and 0 < p 2. Nikolov and Rafailov [20] determined the value M p n for n = 3 and arbitrary p > 0 and also discussed the critical points of U p (ω * n , z) on T . They showed that if p is an even integer with 2 p 2n − 2, then U p (ω * n , z) is constant on T .…”

Section: Planar Case: Equidistributed Setsmentioning

confidence: 99%

“…For equally distributed point sets, it was proven by Stolarsky [27] and by Nikolov and Rafailov [20] that M p (ω * n ) = max z∈T n−1 k=0 |z − ξ k | p is (not necessarily uniquely) attained at z which is, depending on p, either one of the base points ξ k or is the midpoint between two consecutive base points. More precisely, introduce the positive-exponent Riesz energy of ω n ⊂ T defined by E p (ω n ) = n j,k=1 z j − z k p (note that in the previous articles related to Riesz energies, the exponent is taken to be −p, therefore the above quantity becomes the negative exponent Riesz energy).…”

Section: Planar Case: Equidistributed Setsmentioning

confidence: 99%

Polarization, sign sequences and isotropic vector systems

Ambrus

Nietert

2019

Pacific J. Math.

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We determine the order of magnitude of the nth ℓp-polarization constant of the unit sphere S d−1 for every n, d 1 and p > 0. For p = 2, we prove that extremizers are isotropic vector sets, whereas for p = 1, we show that the polarization problem is equivalent to that of maximizing the norm of signed vector sums. Finally, for d = 2, we discuss the optimality of equally spaced configurations on the unit circle.2010 Mathematics Subject Classification. 52A40(primary), and 31C20(secondary).

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“…The p = 4 case of (1.7) along with several higher dimensional generalizations is discussed in the article [9]. A further generalization of the planar problem is given in [17].…”

Section: Proposition 2 the L P Chebyshev Constants Of T Can Be Estimmentioning

confidence: 99%