On the sum of squared distances in the Euclidean plane

Let x(1),...,x(n) be points in the d-dimensional Euclidean space E-d with \\x(i) - x(j)\\ less than or equal to 1 for all 1 less than or equal to i,j less than or equal to n, where \\.\\ denotes the Euclidean norm. We ask for the maximum M(d, n) of Sigma(i,j=1)(n) \\x(i) -x(j)\\(2) (see [4]). This paper deals with the case d = 2. We calculate M(2, n) and show that the value M(2, n) is attained if and only if the points are distributed as evenly as possible among the vertices of a regular triangle of edge-length i. Moreover we give an upper bound for the value Sigma(i,j=1)(n) \\x(i) - x(j)\\, where the points x(1),...,x(n) are chosen under the same constraints as above.