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.

在本文中，我们研究了一个几何不等式猜想，该猜想指出，对于单位半径的半球上的任意四个点，点之间的最大距离之和为 \(4+4\ sqrt{2}\)，即最佳配置是内切于赤道的正正方形，对于任意五个点，最大和为\(5\sqrt{5+2\sqrt{5}}\)最好的配置是内切于赤道的正五边形。我们证明了猜想的配置是局部最优的，并构造了相关可行集中局部极大点的矩形邻域，其大小明确确定，并证明（1）：目标函数受二次多项式约束，其中局部极大点作为邻域中唯一的临界点，以及（2）：可行集的剩余部分可以划分为大量非常小的立方体的有限并，使得在每个小立方体上，猜想可以通过精确的数值计算估计目标函数来验证。