Metrizable in infinite number of ways

Question

The question states that if a topological space is metrizable it is metrizable in infinite number of ways.

Of course scaling the distances by any positive number will do the trick. But i want to know whether any concave and strictly increasing transform applied to distances will also result in a metric.

Edit: Earlier i got confused and asked convex transforms. Thanks @Murthy and Santos for correcting me. The author of this post has proven relatoin between concavity and sub additivity

Answer 1

Partial answer. As pointed out above you cannot have infinite number of different metrics in general. But I will answer the other question you have asked: suppose $f:[0,\infty) \to \mathbb R$ is convex strictly increasing and $f(0)=0$. Can we say that $f(d(x,y))$ is a metric? The answer is no. What you need is not convexity but sub-additivity. For example if $f(x)=x^{2}$ then $f(d(x,y))$ is not a metric since triangle in equality is not satisfied (for example when $d$ is the usual metric on $\mathbb R$).

Answer 2

In general, no, as you have been told. But if you have a metric space in which the metric $d$ takes infinitley many values, then you can consider the family of metrics $\min(d,a)$, with $a>0$. This will give you infinitely many metrics which are equivalent to the original one.