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

作　　者：王玉铮 冷拓 曾振柄[1] WANG Yuzheng;LENG Tuo;ZENG Zhenbing(Department of Mathematics,College of Sciences,Shanghai University,Shanghai 200444,China;School of Computer Engineering and Science,Shanghai University,Shanghai 200444,China)

机构地区：[1]上海大学理学院数学系,上海200444 [2]上海大学计算机工程与科学学院,上海200444

出　　处：《数学年刊（A辑）》2023年第4期409-434,共26页Chinese Annals of Mathematics

基　　金：国家自然科学基金(No.12071282,12171159)的资助.

摘　　要：对于半球面x^(2)+y^(2)+z^(2)=1,≥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。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 x^(2)+y^(2)+z^(2)=1,z≥0(where A,B and C are located on the equator)is 4+4V2.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.

关 键 词：几何不等式 临界点局部分析 分支定界 数学机械化 

分 类 号：O177.92[理学—数学]