Well for

For

a) Any element of S(u;X) has the form Well, to show convexity you have to show that for any So multiplying the p through we get since X is convex, . So we got , then factor out the t to get , which is which is since

b)

Assume X is open.

To show that S(u;X) is open, for any element i need to show that there exists a positive radius r such that this implication holds, if then p must be in S(u;X).

Now, since j is of the form where . Since X is open, there exists a radius such that this implication holds, if then .

Now i will show that for the radius derived from j = tu + x (it is the radius for x, an element of X ), if then .

Now, assume So . So this means that which means Thus , thus which means that S(u;X) is open.