Abstract
In this paper, we study a geometrical inequality conjecture which states that for any four points on a hemisphere with the unit radius, the largest sum of distances between the points is \(4+4\sqrt{2}\), the best configuration is a regular square inscribed to the equator, and for any five points, the largest sum is \(5\sqrt{5+2\sqrt{5}}\) and the best configuration is the regular pentagon inscribed to the equator. We prove that the conjectured configurations are local optimal, and construct a rectangular neighborhood of the local maximum point in the related feasible set, whose size is explicitly determined, and prove that (1): the objective function is bounded by a quadratic polynomial which takes the local maximum point as the unique critical point in the neighborhood, and (2): the remaining part of the feasible set can be partitioned into a finite union of a large number of very small cubes so that on each small cube, the conjecture can be verified by estimating the objective function with exact numerical computation. We also explain the method for constructing the neighborhoods and upper-bound quadratic polynomials in detail and describe the computation process outside the constructed neighborhoods briefly.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Availability of data and materials
the datasets generated during and/or analysed during the current study are available from the corresponding author on request via email.
Code Availability
the Maple code for symbolic computation in Section 3 and Section 4 are available from the corresponding author on request via email, the code for Section 5 are in [17]
References
Fejes-Tóth, L.: On the sum of distances determined by a pointset. Acta Math. Acad. Sci. Hungar. 7, 397–401 (1956)
Stolarsky, K.B.: Sums of distances between points on a sphere II. Proc. Amer. Math. Soc. 41, 575–582 (1973)
Alexander, R.: On the sum of distances betweenn points on a sphere. Acta Math. Acad. Sci. Hungar. 23, 443–448 (1972)
Alexander, R.: On the sum of distances betweenn points on a sphere. II. Acta Mathematica Academiae Scientiarum Hungaricae 29, 317–320 (1977)
Brass, P., Moser, M., Pach, J.: Research Problems in Discrete Geometry. Springer-Verlag, New York (2005)
Berman, J., Hanes, K.: Optimizing the Arrangement of Points on the Unit Sphere. Mathematics of Computation 31(140), 1006–1008 (1977)
Harman, G.: Sums of distances between points on a sphere. Internal. J. Math. 5, 707–714 (1982)
Beck, J.: Some upper bounds in the theory of irregularities of distribution. Acta Arith. 43, 115–130 (1983)
Beck, J.: Sums of distances between points on a sphere ¡ a- an application of the theory of irregularities of distribution to discrete geometry. Mathematika 31(1), 33–41 (1984)
Zirakzadeh, A.: A property of a triangle inscribed in a convex curve. Canad. J. Math. 16, 777–786 (1964)
Bollobás, B.: An extremal problem for polygons inscribed in a convex curve. Canad. J. Math. 19, 523–528 (1967)
Zeng, Z., Zhang, J.: A Mechanical Proof to a Geometric Inequality of Zirakzadeh Through Rectangular Partition of Polyhedra (in Chinese). J Syst Sci Complex 30(11), 1430–1458 (2010)
Hou, Z., Shao, J.: Spherical Distribution of 5 Points with Maximal Distance Sum. Discrete Comput Geom 46, 156–174 (2011)
Zeng, Z., Yang, Z.:How large is the locally optimal region of a locally-optimal point? International Workshop on Certified and Reliable Computation Nanning, Guangxi, China, 17–20, (2011)
Zeng, Z., Xu, Y., Chen, Y., Yang, Z.: A mechanical method for isolating locally optimal points of certain radical functions. F. Boulier et al. (Eds.): CASC 2022, LNCS 13366, pp. 377–396, (2022)
Zeng, Z., Xu, Y., Lu, J. Chen, L.: A Machine Proof of an Inequality for the Sum of Distances between Four Points on the Unit Hemisphere using Maple Software, 2021 Virtual Maple Conference, November 2–5, (2021)
Zeng, Z., Xu, Y., Lu, J., and Chen, L.: Maple Program for MC 2021 Paper: A Machine Proof of an Inequality for the Sum of Distances between Four Points on the Unit Hemisphere using Maple Software, EasyChair Preprints No. 6349
Zeng, Z., Lu, J., Xu, Y., Wang, Y.: Maximizing the Sum of the Distances between Four Points on the Unit Hemisphere. Proceedings of the 13th International Conference on Automated Deduction in Geometry, ADG 2021, Hagenberg, Austria/virtual, September 15-17, EPTCS 352, 27–40. (2021). https://doi.org/10.4204/EPTCS.352.4
Acknowledgements
The authors appreciate very much the suggestions made by the anonymous reviewers on the improvement of this paper, on both English and mathematical expressions. Actually, the proof of Lemma 4 in the current version is given by one of the reviewers
Funding
this work is supported by National Natural Science Foundation of China under Grant Nos. 12171159, 12071282 and 62272416
Author information
Authors and Affiliations
Contributions
the authors of this paper contributed equally to this work
Corresponding author
Ethics declarations
Conflicts of interest
The authors declare that they have no conflict of interest
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Xu, Y., Zeng, Z., Lu, J. et al. Local critical analysis of inequalities related to the sum of distances between n points on the unit hemisphere for \(n=4,5\). Ann Math Artif Intell 91, 865–898 (2023). https://doi.org/10.1007/s10472-023-09880-z
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10472-023-09880-z