Suppose that $u:\mathbb{R}^{n} \to \mathbb{R}$ is a continuous, (weakly) convex function. Now define the value function $\phi$ to be: $$\phi(p,w) = \max_{x>>0} u(x)$$$$ \text{ subject to: } p \cdot x \leq w$$ where $p \in \mathbb{R}^{n}$ and $w \in \mathbb{R}$.
My question is: Under what general conditions will $\phi(p,w)$ be strictly concave in $w$?
I do not know how to approach this problem, but the question/solution has applications in economics and insurance.
It seems to me that first of all, you would need the function $u(x)$ to not attain its optimum in the relative interior. If it did, then changing $w$ would do nothing to your $\phi$.
Your constraint function is linear. If the objective function was also linear, then an increase in $w$ would lead to a linear increase in $\phi$. Problemetic, so we should require strict concavity of $u$.
If $u$ is strictly concave and also strictly increasing, then that should be sufficient conditions.