Consider the following setting. Let $A\in \mathbb{R}^{3 \times m}$ and $B\in \mathbb{R}^{m\times 3}$ be two matrices such that each of their columns must add up to a given $c\in \mathbb{R}$. Denote by $b_1,b_2,b_3$ the columns of $B$. Define the set $\mathcal{B}=\{[b_1,b_2,b_3]: G_1\,b_1\succeq0, G_2\,b_2\succeq 0, G_3\,b_3\succeq 0,\,1^Tb_1 =c,1^Tb_2 =c, 1^T b_3 =c \}$.
For a given $i=1,2,3$, is it possible to formulate the problem below as an LP? $$\max_A \inf_{B\in \mathcal{B}} (AB)_{ii}.$$ I only can express it equivalently as $$\max_A \gamma,$$ subject to $(AB)_{ii} \geqslant \gamma$ for $all$ $B\in\mathcal{B}$. From this, we have an infinite number of inequalities and, what is worse, that it is not linear. Am I right?