Show that a given map does not induce a metric

Question

I have to show that $d(x,y)=\inf|x_i-y_i|$ is not a metric in $\mathbb{R^n}$.

Using the definition $d(x,y)≥0$ is not respected $\forall x,y$ is that enough?

Answer 1

Take $x = (1, 0)$ and $y= (1, 1)$ in $\mathbb{R}^2$. Then $d(x, y) = 0$, but $x \neq y$.

You can easily generalize this to higher dimensions.

Answer 2

This function $d$ is always positive, so this is not the required axiom of a metrics that is in default.

But for example, what si the "distance" between $(1,1)$ and $(1,0)$ with the function $d$ ? Which axiom is not respected for $d$ to be a distance ?

Answer 3

$d(x,y) = \inf\limits_{i=1,\dots,n} |x_i - y_i| \geq 0 \quad \forall x,y \in \mathbb{R}^n$ is indeed satisfied.

Hint: $d(x,y) = 0$ does not imply $x=y$ (and this is one of the properties in the definition of a metric).