1. Suppose that a list contains the values 20 44 48 55 62 66 74 88 93 99 at index positions 0 through 9. Trace the values of the variables….

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

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

2. The class *HSn *of halfspaces over R*n *(see Chapter 9).

3. The class *HHSn *of all homogeneous halfspaces over R*n *(see Chapter 9).

4. The class *Bd *of all functions defined by (open) balls in R*d *. 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 *x*1*, . . ., **xn*}*,** *where for x=(*x*1. *. . ., **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) = 3*x*3* *1 *x*2* *2* *+4*x*3*x*2* *7 is 5).