Given two concentric circles $X_1$ (green), $X_2$ (red) with radii $1$ and $2$, respectively, it is clear that the set of points that maximally separate the two circles in the $L_2$ norm is the circle with radius $\frac{3}{2}$ (blue).

I would like to understand how to construct the set of points that maximally separate $X_1$ and $X_2$ in the $L_1$ norm. I understand the $L_1$ ball generally forms a diamond, but I am somewhat lost trying to figure out how to use this to maximally separate the two circles.

Note: I define the maximum separator as the set of points that have two or more closest points, in the $L_p$ norm, on distinct circles ($X_1$ and $X_2$).