I am having difficulty with the following question. Let $P_n = \{x\in\mathbb{R}^n \; : \; |x_i| \leq 1, i\leq n, \sum_i |x_i| \leq 2 \}$. Build an explicit polyhedral representation of $P_n$ that is linear. That is, a representation of form $P_n = \{x\in\mathbb{R}^n : \exists u\in\mathbb{R}^m \text{ s.t. } Ax + Bu \leq c \}$, where $dim(u)$ and $dim(c)$ are linear in $n$ (also $A, B, c, m$ can depend on $n$).
I have a lot of difficulty with questions asking to building explicit polyhedral representations in general. I think I would see how to do these types of questions in general if I saw how specifically to build a polyhedral representation in this case.
In other words, you want to linearize the absolute values. Introduce variable $u_i$ to represent $|x_i|$. The desired linear constraints in $x$ and $u$ are \begin{align} x_i - u_i &\le 0 \tag1 \\ -x_i - u_i &\le 0 \tag2 \\ u_i &\le 1 \tag3 \\ \sum_i u_i &\le 2 \tag4 \end{align} Constraints $(1)$ and $(2)$ enforce $|x_i| \le u_i$. Constraint $(3)$ enforces $|x_i| \le 1$. Constraint $(4)$ enforces $\sum_i |x_i| \le 2$.