Suppose we are given two $n$-dimensional polyhedra \begin{align} P &= \{ x \in \mathbb{R}^n | \alpha_i(x) \leq a_i, i = 1, \dots, k \} \\ Q &= \{ y \in \mathbb{R}^n | \beta_i(y) \leq b_j, j = 1, \dots, l \} \end{align} where $\alpha_i$ and $\beta_j$ are linear forms and $a_i, b_j \in \mathbb{R}$. I am interested in finding their Minkowski sum $P + Q$.
One way of doing this is to find the corners of $P$ and $Q$, add each corner of $P$ to each corner of $Q$ and take the convex hull of the resulting list of points. (There are some extra steps involved for dealing with the unbounded case, but just the bounded case would be fine for me as well.) I could then translate the result back into the form above, where the polyhedron is given by linear inequalities instead of corners.
However, I have convinced myself geometrically that the set of linear forms that appears in the inequalities describing $P + Q$ will be (or rather, can be chosen to be) a subset of $\{\alpha_1, \dots, \alpha_k, \beta_1, \dots, \beta_l \}$ and the above procedure hides this fact.
Therefore, my question: Is my assumption that these linear forms are enough correct and is there a “direct” way of calculating $P + Q$ that exhibits this?
I assume that this will involve some form of linear optimization, just like calculating the corners would, and I am somewhat hopeful that letting \begin{align} \tilde a_i &= a_i + \min_{y \in Q} \alpha_i(y), \quad i = 1, \dots, k \\ \tilde b_j &= b_j + \min_{x \in P} \beta_j(x), \quad j = 1, \dots, l \end{align} and considering $$ R = \{ z \in \mathbb{R}^n | \alpha_i(z) \leq \tilde a_i, \beta_j(z) \leq \tilde b_j \} $$ might work. Certainly, $P + Q \subseteq R$. However, I can’t see how to get the other inclusion.
A reference on this subject would be sufficient for me as well.