Exercise 4.13: VC Dimension of Spheres
(a)
Every three points that are not colinear can be shattered by half spaces, and, using an epsilon of wiggle room, a half space can be approximated by a sphere by making the radius sufficiently large.
(b)
Consider the convex hull of three points. If the remaining point in inside the convex hull, then, by convexity of a sphere, we cannot not choose that point when choosing the others. Hence, consider the case that the remaining point is outside the convex hull. Then the points can be paired such that drawing lines between the points in a pair makes a plus. The two points corresponding to the longest side cannot be chosen without choosing at least one of the others.