TITLE: Metrics on Euclidean spaces
QUESTION [6 upvotes]: I vaguely remember a theorem that says that any two metrics on the Euclidean space $\mathbb{R}^n$ are equivalent in some sense, but probably not in the sense of metric equivalence: two metrics $d_1$ and $d_2$ are said to be metrically equivalent if there are positive numbers $c_1$ and $c_2$ such that
$c_1 d_1(x,y) \le d_2(x,y) \le c_2 d_1(x,y)$, $\forall$ $x,y \in \mathbb{R}^n$
Can somebody confirm which kind of equivalence there might be? 
Thanks.

REPLY [11 votes]: The correct statement is that any two metrics derived from norms on $\Bbb{R}^n$ are equivalent in this sense. There are plenty of metrics that are not equivalent: for example, the discrete metric is not equivalent to the standard Euclidean metric, but this is because the discrete metric doesn't come from a norm.