For what functions $f$ is $d_f$ defined by $d_f(x,y) = f(d(x,y))$ also a metric on $X$ assuming $d$ is a metric on $X$

Asked Jun 17 '16 at 12:44

Active Jun 17 '16 at 12:44

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For what functions $f : \mathbb{R}_{\leq 0} \to \mathbb{R}_{\leq 0}$ is it true that for every metric $d$ on a set $X$, the function $d_f$ defined by $d_f(x,y) = f(d(x,y))$ is also a metric on $X$?

I know that an example like $f(x) = \sqrt{x}$ works, and any function that works must satisfy $f(x) = 0 \iff x = 0$ but I'd like to know the most general class of functions $f$ that works.

A sub-class of functions would be those that are increasing and satisfy $f(x + y) \leq f(x) + f(y)$ but I don't know if this is a special type of function or not.

asked Jun 17 '16 at 12:44

Anamaki

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The functions you mention are called subadditive. – Giuseppe Negro Jun 17 '16 at 13:01
Oh, thanks! So increasing concave functions with $f(0) = 0$ are some such functions. Are there any others? – Anamaki Jun 17 '16 at 14:21
This has been discussed before on this site, you might want to do a search. – Giuseppe Negro Jun 17 '16 at 15:18
http://math.stackexchange.com/q/507611/8157 – Giuseppe Negro Jun 17 '16 at 15:21

For what functions $f$ is $d_f$ defined by $d_f(x,y) = f(d(x,y))$ also a metric on $X$ assuming $d$ is a metric on $X$

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