Neighbourhood

[Manifold]

Hi!

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

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

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

 

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

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

 

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

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

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

 

Could you please enlighten me on this case? :smile:

[matt grime]

Do you still need an answer? For anyone else wondering you simply draw a picture and think about it for a bit. To be honest the *hardest* to understand is the first one, the one you think is the easiest, by which you simple mean the most familiar