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Sylvester's Four-Point Problem in a Sector and its Relation to Weber Location Problem

This paper obtains the probability that the optimal location of the unweighted Weber location problem is at one of the demand points where demand points are uniformly and independently distributed within a sector. When the number of the demand points is four, this problem is identical with the famous Sylvester’s four-point problem. We solve this problem by use of Crofton’s formula. We have extended the existing works because our results include one of circles and triangles as special cases. Using numerical simulation, we also reveal that the probability is stable unless the angle of the sector is small.