How can the sum of Λth powers (0 < λ < 2) of the Euclidean distances from the variable unit vector p to N fixed unit vectors Pw ,p N be maximized or minimized? By means of an integral transform used in distance geometry, the problem can be reduced in certain cases to minimizing or maximizing sums of integer powers of the inner products (p,Pi). In particular, a complete solution is obtained for the vertices of an m-dimensional octahedron.