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Answering a question about triangle inequality suggested by R. Li, Barrlund [The p-relative distance is a metric. SIAM J Matrix Anal Appl. 1999;21:699.702] introduced a distance function which is a metric on a subdomain of We study this Barrlund metric and give sharp bounds for it in terms of other metrics of current interest. We also prove sharp distortion results for the Barrlund metric under quasiconformal maps.