> 
> Let $C$ be a closed non-empty set, but not necessarily convex. The support function of $C$ is given by
>  $$S(z) = \sup\_{c \in C} \langle z,c\rangle. $$
>  Prove that this is a convex function.
> 
> 
> 

**Attempt**

Let $a \in [0,1]$. Then 

$$S\_C(ax + (1 - a)y) = \sup\_{c \in C} \langle ax + (1- a)y,c\rangle \leq \sup\_{c \in C } \langle a,c\rangle + \sup\_{c \in C} \langle (1-a)y,c\rangle$$

Can I actually break up the sum as $\sup\_{c \in C } \langle a,c\rangle + \sup\_{c \in C} \langle (1-a)y,c\rangle$? If it is not bounded?