Refinement of the mean angle estimation in the Feyesh Toth problem
https://doi.org/10.22405/2226-8383-2022-23-3-245-248
Abstract
The Fejes T´oth problem about the maximum 𝐸* of the mean value of the sum of angles between lines in R3 with a common center is considered. L. Fejes T´oth suggested that 𝐸* = 𝜋/3 = 1.047 . . .. This conjecture has not yet been proven. D. Bilyk and R.W. Matzke proved that 𝐸*<= 1.110 . . .. We refine this estimate using an extremal problem of the Delsarte type: 𝐸* <= 𝐴* <= 1.08326. Using the dual problem 𝐵* we show that the solution of the 𝐴* problem does not allow us to prove the Fejes T´oth conjecture, since 1.05210 < 𝐴*.
Keywords
Fejes T´oth conjecture,
unit sphere,
Legendre polynomial,
linear programming bound,
Delsarte problem.
Supporting agencies: This Research was performed by a grant of Russian Science Foundation (project 18-11-00199), https://rscf.ru/project/18-11-00199/.
About the Authors
Dmitry Viktorovich Gorbachev
Tula State University
Russian Federation
Russian Federation
doctor of physical and mathematical sciences
Daniil Ruslamovich Lepetkov
Tula State University
Russian Federation
Russian Federation
References
1. Andreev, N.N. & Yudin, V.A. 1997. “An extremal location of points on a sphere”, Mat. Pros.,
2. vol. 3, no. 1, pp. 115–125. (In Russ.)
3. Bilyk, D. & Matzke, R.W. 2019. “On the Fejes T´oth problem about the sum of angles between
4. lines”, Proc. Amer. Math. Soc., vol. 147, no. 1, pp. 51–59.
5. Fodor, F., Vıgh, V. & Zarn´ocz, T. 2016. “On the angle sum of lines”, Arch. Math. (Basel),
6. vol. 106, no. 1, pp. 91–100.
Review
For citations:
Gorbachev D.V., Lepetkov D.R. Refinement of the mean angle estimation in the Feyesh Toth problem. Chebyshevskii Sbornik. 2022;23(3):245-248. (In Russ.) https://doi.org/10.22405/2226-8383-2022-23-3-245-248
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