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We were given some examples of metrics in class:

$l$-metric: $d(x,y)= \sum_{i=1}^n |x_i-y_i|$ (open diamonds)

$\infty$-metric: $d(x,y)=\max |x_i-y_i|$ (open squares)

$p$-metric: $d_p(x,y)= \left(\sum_{i=1}^n (x_i-y_i)^p \right)^{1/p}$ (open balls)

and in each case the neighborhoods are shaped differently.

Why do the neighborhoods look like this? How do we look at a metric and determine what the neighborhoods look like?

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    $\begingroup$ Fix a point in your space and look at all the ways to place another point a distance of $1$ away from it. $\endgroup$ – John Douma Nov 5 '15 at 3:05

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