Permutations of finite subsets of R^2 generated by Euclidean distances

Given a finite set of points S = {P1, P2, . . . , Pn} and a vantage point V, generate an ordering of the points of S by measuring the Euclidean distance from V to each of the points of S, ordering them from nearest to farthest. As the vantage point moves around the plane, different orderings will be generated. We are interested in the maximum, minimum, and intermediate values achievable for different point-sets S. (Good and Tideman solved the maximum problem in all dimensions in the 1970s.) We also consider a generalization that uses two vantage points, using the average distance (d(V1, Pk) + d(V2, Pk))/2 to the points of S to generate an ordering.

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