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 p aper deals with the case d = 2. We calculate M(2, n) and show that the valu e M(2, n) is attained if and only if the points are distributed as evenly a s possible among the vertices of a regular triangle of edge-length i. Moreo ver we give an upper bound for the value Sigma(i,j=1)(n) \\x(i) - x(j)\\, w here the points x(1),...,x(n) are chosen under the same constraints as abov e.