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