Optimal measures for $p$-frame energies on spheres

Bilyk, Dmitriy; Glazyrin, Alexey; Matzke, Ryan; Park, Josiah; Vlasiuk, Oleksandr

doi:10.4171/rmi/1329

Cited by 11 publications

(4 citation statements)

References 49 publications

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“…In this case, the cardinality of the support of the global minimizer is therefore exactly d + 1. Furthermore, when Ω = S d with geodesic distance, it is known that tight spherical designs are the unique global minimizers for I F with F (t) = | cos(t)| p , for certain ranges of p [4]. It would be interesting to find other cases where the cardinality of the supports of minimizers can be determined precisely.…”

Section: Discussionmentioning

confidence: 99%

Discreteness of the minimizers of weakly repulsive interaction energies on Riemannian manifolds

Vlasiuk

2020

Preprint

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It is shown that the supports of measures minimizing weakly repulsive energies on Riemannian manifolds with sectional curvature bounded below do not have concentration points. This extends the results of Björck and Carrillo, Figalli, and Patacchini for such energies on the Euclidean space, and complements the results about the discreteness of minimizers of the geodesic Riesz energy on the sphere by Bilyk, Dai, and Matzke.

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Section: Discussionmentioning

confidence: 99%

Discreteness of the minimizers of weakly repulsive interaction energies on Riemannian manifolds

Vlasiuk

2020

Preprint

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“…The p-frame potential has been widely studied (e.g., [5][6][7]). The author has investigated the p-frame potentials of several types of DPPs for the study of finite frames or spherical integrations [8,9].…”

Section: Introductionmentioning

confidence: 99%

On <i>p</i>-frame potentials of the Beltrán and Etayo point processes on the sphere

Hirao

2023

JSIAM Letters

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We address the two novel types of point processes on the multidimensional sphere proposed by Beltrán and Etayo. First, for the point process on the odd-dimensional sphere introduced by Beltrán and Etayo (Constr. Approx., 2018), we give an explicit form of the expectation of the p-frame potential of the process and investigate its asymptotic behavior. Second, for the point processes, termed generalized spherical ensemble, on the even-dimensional sphere introduced by Beltrán and Etayo (J. Math. Anal. Appl., 2019), we provide an upper bound of the expectation of the p-frame potential of the process. Keywords frame potential, projective ensemble, generalized spherical ensemble, determinantal point process, point process on the sphere

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“…When d = 2 this conjecture dates back to Fejes Tóth [12]. For d ≥ 2 it has motivated a recent series of works by Bilyk, Dai, Glazyrin, Matzke, Park, Vlasiuk in different combinations [3] [5] [6] [7], and by Fodor, Vígh and Zarnócz [13]. Other authors have also considered versions of the problem for oriented as well as unoriented lines, in the limit N = ∞ and/or with different powers α of the angle or distance between them, e.g.…”

Section: Introductionmentioning

confidence: 99%

Maximizing expected powers of the angle between pairs of points in projective space

Lim¹,

McCann²

2020

Preprint

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Among probability measures on d-dimensional real projective space, one which maximizes the expected angle arccos( x |x| • y |y| ) between independently drawn projective points x and y was conjectured to equidistribute its mass over the standard Euclidean basis {e 0 , e 1 , . . . , e d } by Fejes Tóth [12]. If true, this conjecture implies the same measure maximizes the expectation of arccos α ( x |x| • y |y|) for any exponent α > 1. For α sufficiently large, we verify the conjecture and establish uniqueness of the resulting maximizer μ up to rotation. In the broader range α ≥ 2, we show μ and its rotations maximize this expectation uniquely on a sufficiently small ball in the L ∞ -Kantorovich-Rubinstein-Wasserstein metric d ∞ from optimal transportation; the same is true for any measure µ which is mutually absolutely continuous with respect to μ, but the size of the ball depends on dµ dμ ∞ .

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Optimal measures for $p$-frame energies on spheres

Cited by 11 publications

References 49 publications

Discreteness of the minimizers of weakly repulsive interaction energies on Riemannian manifolds

Discreteness of the minimizers of weakly repulsive interaction energies on Riemannian manifolds

On <i>p</i>-frame potentials of the Beltrán and Etayo point processes on the sphere

Maximizing expected powers of the angle between pairs of points in projective space

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