Let $f(x)$ be a convex function on $\Pi = [a,b] \times [c,d]$ such that there exists $y \in \mathbb{R}^2$ with property $ y\cdot x\geq f(x), \;\;\; \forall x \in \Pi. $ I want to find such $y$.
Let $x_{j}$, $j=\overline{1,4}$ be corners of $\Pi$. I solve systems $ \left\{ \begin{array}{rcl} y \cdot x_{i_1} & = & f(x_{i_1}), \\ y \cdot x_{i_2} & = & f(x_{i_2}) \end{array}\right. $ for different pairs $(i_1 i_2)$ of corner points and test property $y \cdot x_{i} \geq f(x_{i})$ for two different corner points.
Question 1. Is it true that if these inequalities hold, then $ y \cdot x \geq f(x), \;\;\; \forall x \in \Pi, $ i.e. found $y$ solves my problem?
Question 2. Is it possible that my procedure will not give a solution of my problem?