×

Maximizing expected powers of the angle between pairs of points in projective space. (English) Zbl 1514.90189

The authors consider, as a starting point, the conjecture from 1959 by L. Fejes Tóth [Acta Math. Acad. Sci. Hung. 10, 13–19, I (1959; Zbl 0086.15406)]referring to the fact that the expected angle \(\arccos( \frac{x}{|x|} \cdot \frac{y}{|y|})\) between independently drawn projective points \(x\) and \(y\) equi-distribute its mass over the standard Euclidean basis. As a consequence of the validity of this conjecture it follows that the same measure maximizes the expectation of \(\arccos^{\alpha}(\frac{x}{|x|} \cdot \frac{y}{|y|})\) for any exponent \(\alpha > 1\). The discrete and continuous versions of this conjecture are analyzed in this paper in a non-empty range \(\alpha > \alpha_{\Delta^{d}} \geq 1\). Existence and finiteness of a critical threshold for the exponent \(\alpha\) in various environments, at which the maximizer changes discontinuously are obtained unless the corresponding Fejes Tóth conjecture is true. The uniqueness of the resulting maximizer up to rotation is established. The paper concludes by presenting the continuous version in an Appendix A by Dmitriy Bilyk, Alexey Glazyrin, Ryan Matzke, Josiah Park, and Oleksandr Vlasiuk, in which an effective bound for the critical threshold in the most symmetrical environments is proved.

MSC:

90C26 Nonconvex programming, global optimization
05B30 Other designs, configurations
49Q22 Optimal transportation
52A40 Inequalities and extremum problems involving convexity in convex geometry
58E35 Variational inequalities (global problems) in infinite-dimensional spaces
70G75 Variational methods for problems in mechanics

Citations:

Zbl 0086.15406
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Aigner, M.: Turán’s graph theorem. Am. Math. Mon. 102, 808-816 (1995). doi:10.2307/2974509 · Zbl 0843.05053
[2] Alexander, R.; Stolarsky, KB, Extremal problems of distance geometry related to energy integrals, Trans. Amer. Math. Soc., 193, 1-31 (1974) · Zbl 0293.52005 · doi:10.2307/1996898
[3] Balagué, D.; Carrillo, JA; Laurent, T.; Raoul, G., Dimensionality of local minimizers of the interaction energy, Arch. Ration. Mech. Anal., 209, 1055-1088 (2013) · Zbl 1311.49053 · doi:10.1007/s00205-013-0644-6
[4] Benedetto, JJ; Fickus, M., Finite normalized tight frames, Adv. Comput. Math., 18, 2-4, 357-385 (2003) · Zbl 1028.42022 · doi:10.1023/A:1021323312367
[5] Bilyk, D.; Dai, F., Geodesic distance Riesz energy on the sphere, Trans. Am. Math. Soc., 372, 3141-3166 (2019) · Zbl 1470.11208 · doi:10.1090/tran/7711
[6] Bilyk, D.; Dai, F.; Matzke, R., The Stolarsky principle and energy optimization on the sphere, Constr. Approx., 48, 31-60 (2018) · Zbl 1426.11075 · doi:10.1007/s00365-017-9412-4
[7] Bilyk, D.; Glazyrin, A.; Matzke, R.; Park, J.; Vlasiuk, O., Energy on spheres and discreteness of minimizing measures, J. Funct. Anal., 280, 108995 (2021) · Zbl 1462.31009 · doi:10.1016/j.jfa.2021.108995
[8] Bilyk, D., Glazyrin, A., Matzke, R., Park, J., Vlasiuk, O.: Optimal measures for p-frame energies on the sphere (2019). arXiv:1908.00885 · Zbl 1503.31012
[9] Bilyk, D.; Matzke, RW, On the Fejes Tóth problem about the sum of angles between lines, Proc. Am. Math. Soc., 147, 51-59 (2019) · Zbl 1458.11119 · doi:10.1090/proc/14263
[10] Birkhoff, G., Three observations on linear algebra, Univ. Nac. Tucumán. Revista A, 5, 147-151 (1946) · Zbl 0060.07906
[11] Björck, G., Distributions of positive mass, which maximize a certain generalized energy integral, Ark. Mat., 3, 255-269 (1956) · Zbl 0071.10105 · doi:10.1007/BF02589412
[12] Braides, A., \( \Gamma \)-convergence for Beginners. Oxford Lecture Series in Mathematics and Its Applications (2002), Oxford: Oxford University Press, Oxford · Zbl 1198.49001 · doi:10.1093/acprof:oso/9780198507840.001.0001
[13] Carrillo, JA; Figalli, A.; Patacchini, FS, Geometry of minimizers for the interaction energy with mildly repulsive potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34, 1299-1308 (2017) · Zbl 1408.49035 · doi:10.1016/j.anihpc.2016.10.004
[14] Fejes Tóth, L., Über eine Punktverteilung auf der Kugel, Acta Math. Acad. Sci. Hung., 10, 13-19 (1959) · Zbl 0086.15406 · doi:10.1007/BF02063286
[15] Fodor, F.; Vígh, V.; Zarnócz, T., On the angle sum of lines, Arch. Math. (Basel), 106, 91-100 (2016) · Zbl 1334.51011 · doi:10.1007/s00013-015-0847-1
[16] Kang, K.; Kim, HK; Seo, G., Cardinality estimation of support of the global minimizer for the interaction energy with mildly repulsive potentials, Physica D, 399, 51-57 (2019) · Zbl 1453.49011 · doi:10.1016/j.physd.2019.04.004
[17] Lim, T.; McCann, RJ, Isodiametry, variance and regular simplices from particle interactions, Arch. Ration. Mech. Anal., 241, 553-576 (2021) · Zbl 1470.49038 · doi:10.1007/s00205-021-01632-9
[18] Lim, T., McCann, R.J.: On Fejes Tóth’s conjectured maximizer for the sum of angles between lines. Appl. Math. Optim. 84(3), 3217-3227 (2021). doi:10.1007/s00245-020-09745-5 · Zbl 1478.90073
[19] Lim, T., McCann, R.J.: On the cardinality of sets in \({\mathbf{R}}^d\) obeying a slightly obtuse angle bound. Under revision for SIAM J. Discrete Math · Zbl 07524460
[20] McCann, RJ, Stable rotating binary stars and fluid in a tube, Houston J. Math., 32, 603-632 (2006) · Zbl 1096.85006
[21] Pólya, G.; Szegö, G., Über den transfiniten Durchmesser (Kapazitätskonstante) von ebenen und räumlichen Punktmengen, J. Reine Angew. Math., 165, 4-49 (1931) · JFM 57.0094.03 · doi:10.1515/crll.1931.165.4
[22] Turán, P.: Eine Extremalaufgabe aus der Graphentheorie (Hungarian. German Summary). Mat. Fiz. Lapok 48, 436-452 (1941) · JFM 67.0732.03
[23] Villani, C., Topics in Optimal Transportation. Graduate Studies in Mathematics (2003), Providence: American Mathematical Society, Providence · Zbl 1106.90001
[24] Vlasiuk, O.: Discreteness of the minimizers of weakly repulsive interaction energies on Riemannian manifolds (2020). arXiv:2003.01597
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.