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In this paper, we prove a geometrical inequality which states that for anyfour points on a hemisphere with the unit radius, the largest sum of distancesbetween the points is 4+4*sqrt(2). In our method, we have constructed arectangular neighborhood of the local maximum point in the feasible set, whichsize is explicitly determined, and proved that (1): the objective function isbounded by a quadratic polynomial which takes the local maximum point as theunique critical point in the neighborhood, and (2): the rest part of thefeasible set can be partitioned into a finite union of a large number of verysmall cubes so that on each small cube the conjecture can be verified byestimating the objective function with exact numerical computation.

Maximizing the Sum of the Distances between Four Points on the Unit Hemisphere