How to show that a certain value function is convex?

Question

So I have the following:

Assume $S \subseteq \mathbb{R}^2$ , we have the following function: $v_s: \mathbb{R}^2 \rightarrow \mathbb{R}$ that is defined as follows:

$$\max_{\epsilon_1 , \epsilon_2}(3−2\epsilon_1+\epsilon_2)x_1+ (1 +\epsilon_1−2\epsilon_2)x_ 2 \\ \text{s.t.} (\epsilon_1, \epsilon_2) \in S$$

How can I prove that this function is convex? What steps should I take, because at the moment I do not know what to do with $\epsilon_1$ and $\epsilon_2$

Answer 1

Note that for any $\epsilon_1,\epsilon_2\in S$, $$f_{\epsilon_1,\epsilon_2}(x_1,x_2)\to(3−2\epsilon_1+\epsilon_2)x_1+ (1 +\epsilon_1−2\epsilon_2)x_ 2$$ is convex because it is linear. Therefore $$v_s(x_1,x_2)=\max_{\epsilon_1,\epsilon_2\in S}f_{\epsilon_1,\epsilon_2}(x_1,x_2)$$ is still convex because Proving that the maximum of two convex functions is also convex