In this paper it is shown that the diameter D(P) of a set of n points P on the plane is not necessarily an edge in the dual of the furthest-point Voronoi diagram (FPVD) of P, as previously claimed in  and . It is also proved that if P is contained in the disk determined by D(P) then the above property does hold. Furthermore, it is shown that an edge e in the dual of the FPVD(P) intersects its corresponding edge in the FPVD(P) if, and only if, P is contained in the disk determined by e. These results invalidate several algorithms for solving the diameter, all-furthest-neighbor, and maximal spanning tree problems proposed in  and . A proof of correctness is given for the minimum spanning circle algorithm proposed in  and . Finally new O(n log n) algorithms are offered for the minimum spanning circle and all-furthest-neighbor problems.