Hi I'm having trouble picturing the open balls for the Railway metric
(,)={2(,) if ,,0 are collinear
2(,0)+2(0,) otherwise
I need to sketch the open balls of Bd(0,1), Bd((1,0),1) and Bd((1/2,0),1)
I have seen pictures of them like a "lollipop" but don't know how to show for these cases.
The name "Railway Metric" comes from the following image. Suppose there are a number of railway lines which start at the origin $O$ and go radially outwards in straight lines. From a station $A$ on line $OA$ you can travel directly to any other point on the line $OA$ towards $O$ or away from $O$, but to reach a station on another line you have travel in to $O$, change to the other line, and then travel back out again.
This gives you a useful intuitive picture of how this metric behaves.
To picture the open ball with radius $1$ around $O$ ask yourself the following question "what places can I travel to in less than $1$ hour by train if I start at $O$ ?".
To picture the open ball with radius $1$ around $(1,0)$ ask yourself the following question "what places can I travel to in less than $1$ hour if I start at $A$ which is $1$ hour by train from $O$ ?".
To picture the open ball with radius $1$ around $(\frac 1 2,0)$ ask yourself the following question "what places can I travel to in less than $1$ hour if I start at $B$ which is $\frac 1 2$ hour by train from $O$ ?".