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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.

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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 $O$ ?".

gandalf61
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  • Thank you so much. This is exactly the kind of explanation I was looking for! –  Oct 09 '19 at 13:16
  • could i confirm my answers now with you. For Bd(0,1) I would have a circle around the origin, radius 1. For Bd((1,0),1) I would have a Line from 1 to the origin and a ball around the origin (is the radius of this ball now 1/2?) For Bd((1/2,0),1) would this also be a ball and line, this time values wrt (1/2,0) –  Oct 10 '19 at 10:31
  • @mathsstudent101 For $Bd((1,0),1)$ and $Bd((\frac 1 2,0),1)$ don't forget that you can travel away from the origin as well as towards it. – gandalf61 Oct 10 '19 at 10:54
  • But if you travel away too, taking (1,0) as eg you would not be within the hour train from O? –  Oct 10 '19 at 11:13
  • @mathsstudent101 $Bd((1,0),1)$ is the open ball of radius $1$ around $(1,0)$ - so in the train analogy you can travel up to 1 hour starting from $(1,0)$, not starting from the origin. – gandalf61 Oct 10 '19 at 11:22