1. Show that each of the following classes can be represented as a Dudley class:

2. The class HSn of halfspaces over Rn (see Chapter 9).

3. The class HHSn of all homogeneous halfspaces over Rn (see Chapter 9).

4. The class Bd of all functions defined by (open) balls in Rd . Use the Dudley representation to figure out the VC-dimension of this class.

5. Let Pd n denote the class of functions defined by polynomial inequalities of degree ≤ d, namely, Pd n = {h p : p is a polynomial of degree ≤ d in the variables x1, . . ., xn}, where for x=(x1. . . ., xn), h p(x)=1[p(x)≥0] (the degree of amultivariable polynomial is the maximal sum of variable exponents over all of its terms. For example, the degree of p(x) = 3x3 1 x2 2 +4x3x2 7 is 5).