On the Fejes Tóth problem about the sum of angles between lines
Authors:
Dmitriy Bilyk and Ryan W. Matzke
Journal:
Proc. Amer. Math. Soc. 147 (2019), 51-59
MSC (2010):
Primary 11K38, 52C35
DOI:
https://doi.org/10.1090/proc/14263
Published electronically:
October 18, 2018
MathSciNet review:
3876730
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: In 1959 Fejes Tóth posed a conjecture that the sum of pairwise nonobtuse angles between unit vectors in
is maximized by periodically repeated elements of the standard orthonormal basis. We obtain new improved upper bounds for this sum, as well as for the corresponding energy integral. We also provide several new approaches to the only settled case of the conjecture:
.
- [1] John J. Benedetto and Matthew Fickus, Finite normalized tight frames, Adv. Comput. Math. 18 (2003), no. 2-4, 357–385. Frames. MR 1968126, https://doi.org/10.1023/A:1021323312367
- [2] D. Bilyk and F. Dai, Geodesic Riesz Energy on the Sphere (2017), arXiv:1612.08442.
- [3] Dmitriy Bilyk, Feng Dai, and Ryan Matzke, The Stolarsky principle and energy optimization on the sphere, Constr. Approx. 48 (2018), no. 1, 31–60. MR 3825946, https://doi.org/10.1007/s00365-017-9412-4
- [4] Dmitriĭ Bilik and Maĭkl T. Lèĭsi, One bit sensing, discrepancy, and the Stolarsky principle, Mat. Sb. 208 (2017), no. 6, 4–25 (Russian, with Russian summary); English transl., Sb. Math. 208 (2017), no. 5-6, 744–763. MR 3659577, https://doi.org/10.4213/sm8656
- [5] S. Borodachov, D. Hardin, and E. Saff, Minimal Discrete Energy on Rectifiable Sets, Springer, Monographs in Math. (to appear).
- [6] M. Ehler and K. A. Okoudjou, Minimization of the probabilistic 𝑝-frame potential, J. Statist. Plann. Inference 142 (2012), no. 3, 645–659. MR 2853573, https://doi.org/10.1016/j.jspi.2011.09.001
- [7] L. Fejes Tóth, Über eine Punktverteilung auf der Kugel, Acta Math. Acad. Sci. Hungar. 10 (1959), 13–19 (unbound insert) (German, with Russian summary). MR 105654, https://doi.org/10.1007/BF02063286
- [8] F. Fodor, V. Vígh, and T. Zarnócz, On the angle sum of lines, Arch. Math. (Basel) 106 (2016), no. 1, 91–100. MR 3451371, https://doi.org/10.1007/s00013-015-0847-1
- [9]
F. Petrov (https://mathoverflow.net/users/4312/fedor-petrov), Maximum sum of angles between
lines, URL (version: 2014-07-09): https://mathoverflow.net/q/173712 .
- [10] Kenneth B. Stolarsky, Sums of distances between points on a sphere. II, Proc. Amer. Math. Soc. 41 (1973), 575–582. MR 333995, https://doi.org/10.1090/S0002-9939-1973-0333995-9
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Additional Information
Dmitriy Bilyk
Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55408
Email:
dbilyk@math.umn.edu
Ryan W. Matzke
Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55408
Email:
matzk053@umn.edu
DOI:
https://doi.org/10.1090/proc/14263
Received by editor(s):
January 23, 2018
Published electronically:
October 18, 2018
Additional Notes:
This work is supported by the NSF grant DMS 1665007 (the first author) and the NSF Graduate Research Fellowship 00039202 (the second author)
Communicated by:
Alexander Iosevich
Article copyright:
© Copyright 2018
American Mathematical Society