Local critical analysis of inequalities related to the sum of distances between n points on the unit hemisphere for \(n=4,5\)

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.