This question is related my previous question (Comparing two probability distributions) which are both related to my current research.
Suppose we have two bounded convex hulls in $\mathbb{R}^n$ defined by the two sets of linear inequalities $A_1x \geq 0$ and $A_2x \geq 0$ and the common equality $\sum x=a$ where $A_1$, $A_2$ are real valued matrices, $a$ is a real positive constant and $x \in \mathbb{R}^n$. What conditions must be satisfied by $A_1$, $A_2$ for the first convex hull to be contained in the second convex hull?
Thank you.
Let's assume $\{x: A_1 x \ge 0, \sum x = a\}$ is nonempty. For each row $R$ of $A_2$, you want every solution of $A_1 x \ge 0$, $\sum x = a$ to satisfy $R \cdot x \ge 0$, i.e. the optimal value of the linear programming problem P:
minimize $R \cdot x$ subject to $A_1 x \ge 0$, $\sum x = a$
is at least $0$. The dual of this linear programming problem is D:
maximize $a z$ subject to $ A_1^T y + z {\bf 1} = R$, $y \ge 0$ (where $\bf 1$ is the vector of all $1$'s)
By duality, P has a solution with objective value $ < 0$ if and only if D has no solution with objective value $\ge 0$. Your condition is equivalent to: there exist $y \ge 0$ and $z\ge 0$ such that $R = A_1^T y + z \bf 1$, i.e. each row of $A_2$ is a linear combination with nonnegative coefficients of $\bf 1$ and the rows of $A_1$.