Minimization of a sum of taxicab distances formulated as a linear program 
Suppose we are given $n$ points $A_1, \dots, A_n \in \mathbb{R}^2$. The task is to find a point $x = (x_1,x_2) \in \mathbb{R}^2$ such that the sum of distances to the points $A_1, \dots, A_n$ in the $\ell_1$-norm is minimized. Formulate this problem as a linear program.

So, first of all, the $\ell_1$-norm of a point $x=(x_1,x_2)\in\mathbb{R}^2$ is $\|x\|_1=|x_1|+|x_2|$. The problem would then be
$$\min \sum\limits_{i=1}^n \|x-A_i\|_1 = \min \sum\limits_{i=1}^n|x_1-A_{i1}|+|x_2-A_{i2}|$$
with no constraints. But how can one formulate this as a linear program?

version 1.
$$\begin{align}
  \min & \sum_{i,j} y^+_{i,j} + y^-_{i,j}\\
  &y^+_{i,j} - y^-_{i,j} = x_j-A_{i,j}\\
  &y^+_{i,j}\ge 0, y^-_{i,j}\ge 0
\end{align}$$
version 2.
$$\begin{align}
  \min & \sum_{i,j} y_{i,j}\\
  &-y_{i,j} \le x_j-A_{i,j} \le y_{i,j}\\
  &y_{i,j}\ge 0
\end{align}$$
