Functions transferring metrics to metrics

We study the properties of real functions f for which the compositions f ◦ d is a metric for every metric space (X, d). The explicit form is found for the invertible elements of the semigroup $${\mathcal F}$$ of all such functions. The increasing functions $${f \in \mathcal F}$$ are characterized by the subadditivity condition and a maximal inverse subsemigroup in the set of these functions is explicitly described. The upper envelope of the set of functions $${f \in \mathcal F}$$ with f (1) = 1 is found and it leads to the exact constant in Harnack’s inequality for functions from $${\mathcal F}$$ .