The authors consider three \(p\)-dimensional jointly distributed random vectors \(X,Y\) and \(Z\) with respective normal marginal distributions \(N(0,\Sigma_{ii})\), \(i=1,2,3\) and determine certain covariance matrices that minimize the sum of the \(L_ 2\)-distances of the three vectors. This problem posed in a statistical context is equivalent to maximizing submatrix traces of a positive definite matrix.