Problem 2.3 c
the last comment confused me a little bit.
For a given set of N points, we should change the center of the sphere to get as many dichotomies as we can, thus measuring the effective number of hypotheses (spheres) in this hypothesis set.
Does it make sense to move project the spheres from 3D to 1D and look at the problem as intervals of +1 for a<=x<=b and a>=x>=b?
