On the Partition Cells Formed by Hyperspheres

This paper deals with the partition cells formed by m hyperspheres in an n-dimensional real space Rⁿ. Such partition cells are in a direct correspondence to single threshold or sharp translated decision functions which appear in the theory of neural networks and fuzzy systems. Both the least upper bound and the greatest lower bound on the number of the partition cells formed by the hyperspheres are derived and formulated. Compared with the bounds on the number of partition cells formed by the same number of hyperplanes, partition cells formed by the hyperspheres could be much more complex than those by the hyperplanes when m is much greater than n, whereas the decision functions corresponding to the hyperspheres may have the same number of parameters as those to the hyperplanes.