Abstract
Choose N unoriented lines through the origin of \({\mathbf{R}}^{d+1}\). The sum of the angles between these lines is conjectured to be maximized if the lines are distributed as evenly as possible amongst the coordinate axes of some orthonormal basis for \({\mathbf{R}}^{d+1}\). For \(d \ge 2\) we embed the conjecture into a one-parameter family of problems, in which we seek to maximize the sum of the \(\alpha \)th power of the renormalized angles between the lines. We show the conjecture is equivalent to this same configuration becoming the unique optimizer (up to rotations) for all \(\alpha >1\). We establish both the asserted optimality and uniqueness in the limiting case \(\alpha =\infty \) of mildest repulsion. The same conclusions extend to \(N=\infty \), provided we assume only finitely many of the lines are distinct.
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TL is grateful for the support of ShanghaiTech University, and in addition, to the University of Toronto and its Fields Institute for the Mathematical Sciences, where parts of this work were performed. RM acknowledges partial support of his research by Natural Sciences and Engineering Research Council of Canada Grants RGPIN-2015-04383 and 2020-04162. ©2020 by the authors.
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Lim, T., McCann, R.J. On Fejes Tóth’s Conjectured Maximizer for the Sum of Angles Between Lines. Appl Math Optim 84, 3217–3227 (2021). https://doi.org/10.1007/s00245-020-09745-5
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DOI: https://doi.org/10.1007/s00245-020-09745-5
Keywords
- Potential energy minimization
- Spherical designs
- Projective space
- Extremal problems of distance geometry
- Great circle distance
- Attractive-repulsive potentials
- Mild repulsion limit
- Riesz energy