Draw the ball B((x,y), 1) of Jungle River metric

Question

Draw the ball $B((x,y), 1)$ of Jungle River metric and describe this set.

I know the answer for the first case, but I am not sure about the second. I thought that it is a diamond but when I tried to prove that I became less sure. The set must be describe for every $(x,y)$.

please also see here: https://math.stackexchange.com/questions/662616/help-sketching-jungle-river-metric-in-mathbbr2 — Rahul Madhavan, Apr 08 '21 at 20:20
In General Topology by R. Engelking this is called the River Metric; he says vertical lines are separated by mountains. — DanielWainfleet, Apr 09 '21 at 06:52

score 2 · Accepted Answer · answered Apr 08 '21 at 20:29

I’ll summarize what the open unit balls look like at different points and leave it to you to verify that the descriptions are correct.

It will depend on $|y|$. If $|y|\ge 1$, then of course it is simply $\{x\}\times(y-1,y+1)$. If $y=0$, it’s an open square diamond with vertices $\langle x-1,0\rangle$, $\langle x+1,0\rangle$, $\langle x,1\rangle$, and $\langle x,-1\rangle$. And if $0<|y|<1$, it’s a similar, smaller diamond with a vertical ‘tail’. The centre of the diamond is at $\langle x,0\rangle$, the vertices are at distance $1-|y|$ from the centre, and the tail is the part of $\{x\}\times(y-1,y+1)$ that sticks out of the diamond.

Draw the ball B((x,y), 1) of Jungle River metric

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