Sphere-separable partitions of multi-parameter elements

We show the optimality of sphere-separable partitions for problems where nn vectors in dd-dimensional space are to be partitioned into pp categories to minimize a cost function which is dependent in the sum of the vectors in each category; the sum of the squares of their Euclidean norms; and the number of elements in each category. We further show that the number of these partitions is polynomial in nn. These results broaden the class of partition problems for which an optimal solution is guaranteed within a prescribed set whose size is polynomially bounded in nn. Applications of the results are demonstrated through examples.