The second-order condition tells us that a twice differentiable is

*strictly*convex if has for all , but not vice versa. Here we verify two examples.- Show that is a strictly convex function.
- Show that is a strictly convex function, and there exists such that (i.e., there exists such that ).

For each of the following functions determine whether it is convex, concave, or neither.

- on .
- on .
- on , where .

We've showed that any norm is a convex function. In particular, for all , -norm defined by is a convex function on .

Now we consider the case where . Is the function convex or concave, or neither? Prove your conclusion.

Let on , where is a fixed integer, denotes the vector with , and is the -th largest component of . In other words, are the absolute values of the components of , sorted in nonincreasing order.

Show that is a convex function.

(Hint: Express as the pointwise maximum of a family of convex functions.)