Why is $d(x,0)$ not a norm? If $\|x\|$ is a norm, then we can define $d(x,y):=\|x-y\|$ and it will be a metric. Now, if $d$ is a metric, why is $\|x\|:= d(x,0)$ not a norm? I think it fail for the sub-linearity, but I don't see how.

Take for example $d$ as the discrete distance, then for $x\not=0$ and $|\lambda|\not=0,1$, we
$$1=d(\lambda x,0)=\|\lambda x\|\not=|\lambda|\|x\|=|\lambda|d(x,0)=|\lambda|$$
and the absolute homogeneity property does not hold.
