Maximizing the Sum of the Distances between Four Points on the Unit Hemisphere

Zhenbing Zeng
(Shanghai University, Department of Mathematics, Shanghai 200444, China)
Jian Lu
(Shanghai University, Department of Mathematics, Shanghai 200444, China)
Yaochen Xu
(Shanghai University, Department of Mathematics, Shanghai 200444, China)
Yuzheng Wang
(Shanghai University, Department of Mathematics, Shanghai 200444, China)

In this paper, we prove a geometrical inequality 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). In our method, we have constructed a rectangular neighborhood of the local maximum point in the feasible set, which size is explicitly determined, and proved 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 rest 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.

In Predrag Janičić and Zoltán Kovács: Proceedings of the 13th International Conference on Automated Deduction in Geometry (ADG 2021), Hagenberg, Austria/virtual, September 15-17, 2021, Electronic Proceedings in Theoretical Computer Science 352, pp. 27–40.
Published: 30th December 2021.

ArXived at: http://dx.doi.org/10.4204/EPTCS.352.4 bibtex PDF
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