Proper gromov transforms of metrics are metrics

In phylogenetic analysis, a standard problem is to approximate a given metric by an additive metric. Here it is shown that, given a metric D defined on some finite set X and a non-expansive map f : X → R, the one-parameter family of the Gromov transforms D of D relative to f and ∆ that starts with D for large values of ∆ and ends with an additive metric for ∆ = 0 consists exclusively of metrics. It is expected that this result will help to better understand some standard tree reconstruction procedures considered in phylogenetic analyis.