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I've got a question concerning neighbourhoods of points in 2- and 3-dimensional space.

How can we explicitly show, using only the definition of a given metric, that the according neighbourhood is some figure?

For example the three metrics are given:

1) [itex]d(x,y)=\sqrt{\sum_{i=1}^{n}{(x_i-y_i)^2}}[/itex];

2) [itex]d(x,y)=\displaystyle\max_i|x_i-y_i|[/itex];

3) [itex]d(x,y)=\sum_{i=1}^{n}|x_i-y_i|[/itex]

If [itex]a\in{\mathbb{R}}^2[/itex] and the metric is 1), then it's clear:

[itex]d(a,x)=\sqrt{(a_1-x_1)^2+(a_2-x_2)^2}<\epsilon[/itex] and it's clear that it's a circle...because one can rewrite:[itex](a_1-x_1)^2+(a_2-x_2)^2<{\epsilon}^2[/itex]

But I have problems to see a picture when looking at the other metrics:

[itex]d(a,x)=\displaystyle\max_2\{|a_1-x_1|,|a_2-x_2|\}<\epsilon[/itex];

[itex]d(a,x)=|a_1-x_1|+|a_2-x_2|<\epsilon[/itex] tell me nothing at the moment about what the according neighbourhood might look like.

Could you please enlighten me on this case?

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# Points in 2- and 3-dimensional space

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