Absolute minima of potentials of certain regular spherical configurations

Borodachov, Sergiy V.

doi:10.1016/j.jat.2023.105930

Journal of Approximation Theory

2023

DOI: 10.1016/j.jat.2023.105930

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Absolute minima of potentials of certain regular spherical configurations

Sergiy V. Borodachov

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2024

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Cited by 2 publications

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Odd strength spherical designs attaining the Fazekas–Levenshtein bound for covering and universal minima of potentials

Borodachov

2024

Aequat. Math.

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We characterize the cases of existence of spherical designs of an odd strength attaining the Fazekas–Levenshtein bound for covering and prove some of their properties. We also find all universal minima of the potential of regular spherical configurations in two new cases: the demihypercube on $$S^d$$ S d , $$d\ge 4$$ d ≥ 4 , and the $$2_{41}$$ 2 41 polytope on $$S^7$$ S 7 (which is dual to the $$E_8$$ E 8 lattice).

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Odd strength spherical designs attaining the Fazekas–Levenshtein bound for covering and universal minima of potentials

Borodachov

2024

Aequat. Math.

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Bounds on Polarization Problems on Compact Sets via Mixed Integer Programming

Rolfes,

Schüler,

Zimmermann

2024

Discrete Comput Geom

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Finding point configurations, that yield the maximum polarization (Chebyshev constant) is gaining interest in the field of geometric optimization. In the present article, we study the problem of unconstrained maximum polarization on compact sets. In particular, we discuss necessary conditions for local optimality, such as that a locally optimal configuration is always contained in the convex hull of the respective darkest points. Building on this, we propose two sequences of mixed-integer linear programs in order to compute lower and upper bounds on the maximal polarization, where the lower bound is constructive. Moreover, we prove the convergence of these sequences towards the maximal polarization.

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Absolute minima of potentials of certain regular spherical configurations

Cited by 2 publications

References 13 publications

Odd strength spherical designs attaining the Fazekas–Levenshtein bound for covering and universal minima of potentials

Odd strength spherical designs attaining the Fazekas–Levenshtein bound for covering and universal minima of potentials

Bounds on Polarization Problems on Compact Sets via Mixed Integer Programming

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