Consider $n$ convex polytopes $S_1, \cdots, S_n$ and a set of matrices $W$ such that each matrix $A\in W$, we have that the $i$-th row of $A$ is a member of $S_i$. (In general $W$ is infinite.)
Furthermore, given a vector $e$, I want a finite set $F$ of vectors from $W$ and $e$, such that for any $x,y$
$\{xAe \mid A\in W\} = \{yAe \mid A\in W\}$ iff $x \cdot f =y\cdot f$ for each $f\in F$.
Is there a simple way to come up with such an $F$?