How can the sum of λ-th
powers (0 < λ < 2) of the Euclidean distances from the variable unit vector p to N
fixed unit vectors p1,⋯,pN 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.