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 nonexpansive map f : X --> R, the one-parameter family of the Gromov transforms D-Delta,D-f of D relative to f and A that starts with D for large values of A and ends with an additive metric for Delta = 0 consists exclusively of metrics. It is expected that this result will help to better understand some standard tree reconstruction procedures considered in phylogenetic analysis. (C) 2002 Elsevier Science Ltd. All rights reserved.