On extremal point distributions in the Euclidean plane

We ask for the maximum sigma(n)(gamma) of Sigma(i,j=1)(n) parallel tox(i)-x(j)parallel to(gamma), where x(1),...,x(n) are points in the Euclidean plane R-2 with parallel tox(i) - x(j)parallel to less than or equal to 1 for all 1 less than or equal to i, j less than or equal to n and where parallel to.parallel to(gamma) denotes the gamma-th power of the Euclidean norm, gamma greater than or equal to 1. (For gamma = 1 this question was stated by L. Fejes Toth in [1].) We calculate the exact value of sigma(n)(gamma) for all gamma greater than or equal to 1,0758... and give the distributions which attain the maximum sigma(n)(gamma). Moreover we prove upper bounds for sigma(n)(gamma) for all gamma greater than or equal to 1 and calculate the exact value of sigma(4)(gamma) for all gamma greater than or equal to 1.