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Please could someone show me an example of metrics $d$ and $d'$ that are not strongly equivalent but are equivalent?

I read the Wikipedia article here but couldn't find an example. For completeness let me reproduce the definitions here:

Two metrics $d,d' : X \times X \to \mathbb R$ are called equivalent if for every $x_0 \in X$ and every $r > 0$ there exist $r' > 0$ such that

$$ B_d(x_0, r') \subseteq B_{d'}(x_0, r)$$ and $r'' > 0$ such that $$ B_{d'}(x_0, r'') \subseteq B_{d}(x_0, r)$$

Two metrics $d,d'$ are called strongly equivalent if there exist constants $c,C>0$ such that for all $x,y \in X$:

$$ c d(x,y) \le d'(x,y) \le C d(x,y)$$

a student
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1 Answers1

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Let $X = \mathbb{R}^n$ and $d$ the usual euclidean metric. Define $d'(x,y) = \frac{d(x,y)}{d(x,y) + 1}$ for $x,y \in \mathbb{R}^n$. As the map $\mathbb{R}_{\geq 0} \rightarrow \mathbb{R}_{\geq 0}, x \mapsto \frac{x}{x + 1}$ is monotone and concave, it can be shown that $d'$ is a metric on $X$.

Notice that $B_{d'}(x_0,r) = X$ for all $r \geq 1$. We thus have $B_d(x_0,r') \subseteq B_{d'}(x_0,r)$ with $r' = \frac{r}{1-r}$ if $r < 1$ and $r' = 1$ if $r \geq 1$. On the other hand we have $d' \leq d$, so $B_{d}(x_0,r) \subseteq B_{d'}(x_0,r)$ for all $r > 0$.

Finally note that $cd \leq d'$ for some $c > 0$ is equivalent to $ c \leq \frac{1}{d + 1}$, but taking some $x,y \in X$ with large distance we see that this inequality does not hold for any $c > 0$.

Matthias Klupsch
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