I'll just post because I don't think Eli Lansey’s answer uses the right definition for post office metric. I like the other name of the post office metric better: British rail metric. It assumes that, when going from point A to point B, the fastest path is to go via London (i.e., the origin), unless of course you're already at your destination!
So, we consider a fixed point $\mathbf{p}$, the ball $B_r(\mathbf{p})$ is the set of points $\mathbf{q}$ that satisfy:
$$\|\mathbf{p}\|^2 + \|\mathbf{q}\|^2 < r^2$$
that is, if we have $\mathbf{q}=(x,y)$, the ball $B_r(\mathbf{p})$ is the union of the $\{\mathbf{p}\}$ and all points satisfying:
$$x^2 + y^2 < r^2 - \|\mathbf{p}\|^2$$
The latter is the ball of radius $r' = \sqrt{r^2 - \|\mathbf{p}\|^2}$ around the origin for the Euclidean distance in the plane, which we might note $B_{r'}^{\text{E}}(\mathbf{O})$. To summarize:
- if $r < \|\mathbf{p}\|$, $B_r(\mathbf{p}) = \{\mathbf{p}\}$
- otherwise, $B_r(\mathbf{p}) = \{\mathbf{p}\} \cup B_{r'}^{\text{E}}(\mathbf{O})$ with $r' = \sqrt{r^2 - \|\mathbf{p}\|^2}$
Okay, this being Mathematica.SE, I figure I could give code to draw the above, in addition to doing the maths. So, this draws the ball (point $\mathbf p$, which is part of the ball, is drawn as a little filled square so it's visible):
ball[p_, r_] := Show[
RegionPlot[
x^2 + y^2 + p[[1]]^2 + p[[2]]^2 < r^2, {x, -5, 5}, {y, -5, 5},
PerformanceGoal -> "Quality"],
Graphics[{Blue, Point[p]}]
]
and this is an animation of a ball of radius 3 as its center $\mathbf p$ moves from $(0,0)$ to $(0,4)$: