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.
翻译:在本文中,我们证明存在着几何不平等,它表明,对于具有单位半径的半球的任何四点,两点之间的最大距离总和是 4+4*sqrt(2)。用我们的方法,我们已经在可行的成套标准中建造了一个当地最大点的矩形邻区,其大小已明确确定,并证明:(1) 目标功能受四边多面体的结合,该四边体体体将当地最大点作为邻里的独特临界点;(2) 可行组合的其余部分可以分割成一大批非常小的立方体的有限结合,这样每个小立方体的预测可以通过精确的数值计算来估计客观函数来核实。