Condition for value function to be strictly concave

Question

Consider the optimization problem $$ c(p) = \min_x \sum_{i=1}^n x_ip_i $$ subject to $f(x)\geq 1$ where $f:\mathbb{R}^n_+\mapsto \mathbb{R}$ is increasing and concave. I know that $c$ is concave (see here for a proof). My question is: are there conditions on $f$ such that $c$ is strictly concave? Many thanks for any help!

Answer 1

Notice that since $f$ is increasing, $p_i \geq 0$ for all $i$; otherwise we do not have the solution for problem. Then at the optimal solution $x(p)$, we must have $f(x(p))=1$. Using this remark, one sufficient condition for $c(p)$ is strictly concave is that $f$ is strictly concave.