Convexity of the value function consider the following

Question

Question: Prove Theorem.

Theorem: Convexity of the value function. Consider the following problem and the associated value function:

v(α) = max{f(x; α); g(x) ≥ 0}

where α ε Ω a convex set. (Note that the parameters do not enter the constraint function.) If the objective function is convex in the parameters a for any given x, then V( ) is convex.