For the value iteration function, if the immediate cost is a linear function in state, then is this value iteration function convex?

Question

For the state value function $V(s)$, if the immediate cost is a linear function in the state, for example $r(s,a) = s + a^2$, where $s$ and $a$ are state and action, which are finite. Is this value function convex in state $s$? if yes, how to prove it?

BTW, the Bellman optimal equation is: $V(s) = \min_{a\in \mathcal{A}} \Big\{r(s,a) + \gamma \sum_{s'\in \mathcal{S}} p(s'|s,a) V(s') \Big\}$. $\gamma \in [0,1]$ is the discount factor.

Answer 1

Possible duplicate: Is linear function convex or concave?

This link should answer your question. If the value function is linear in $s$, it is also convex (and concave!) in $s$.