| 王玉铮,冷拓,曾振柄.半球面上四点距离之和的最大值问题*[J].数学年刊A辑,2023,44(4):409~434 |
| 半球面上四点距离之和的最大值问题* |
| On the Maximum of the Sum of the Distances of Four Points on the Hemisphere |
| Received:December 06, 2021 Revised:June 12, 2023 |
| DOI：10.16205/j.cnki.cama.2023.0029 |
| 中文关键词: 几何不等式, 临界点局部分析, 分支定界, 数学机械化 |
| 英文关键词:Geometric inequality, Local analysis of critical points, Branch and bond, Mathematics mechanization |
| 基金项目:国家自然科学基金(No.12071282, 12171159) |
| Author Name | Affiliation | | WANG Yuzheng | Department of Mathematics, College of Sciences, Shanghai University, Shanghai 200444, China. | | LENG Tuo | School of Computer Engineering and Science, Shanghai University, Shanghai 200444, China. | | ZENG Zhenbing | Corresponding author. Department of Mathematics, College of Sciences,Shanghai University, Shanghai 200444, China. |
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| 中文摘要: |
| 对于半球面x2 + y2 + z2 =1,z ≧ 0上的四点 A,B,C,D ,其中 A,B,C 三点位于赤道上,本文证明了它们两两之间距离和的最大值为 4+4√2 .由于距离求和计算中存在根号相加, 无法直接用通常的微积分方法处理,因此, 本文将问题分成二步进行证明. 首先, 在一个局部极大值点的附近构造了一个小邻域，通过估计一个级数的展开式证明了此定理在这个小邻域上成立,然后在该邻域外通过分支定界和计算机数值计算, 证明了此定理成立. |
| 英文摘要: |
| In this paper, the anthors proved that the maximum sum of the distances between the four points A, B, C and D on the hemisphere x2 + y2 + z2 = 1, z > 0 (where A, B and C are located on the equator) is 4 + 4√2. This problem is hard for pure symbolic deduction because sum of several square root is involved in the distance calculation. The anthors present a two-stage method to solve the inequality. In the first stage, the authors constructed a small cube around the local maximal point and proved that in the cubic neighbourhood the local maximum is also the global maximum by estimating the series expansion of the object function, and in the second stage the anthors verified the correctness of the inequality outside the neighorhood by using the branch and bound method and Python scientific computing. |
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