Barrlund's distance function

Answering a question about triangle inequality suggested by R. Li, A. Barrlund introduced a distance function which is a metric on a subdomain of ${\mathbb R}^n\,.$ P. H\"ast\"o considerably extended Barrlund's results. We study this function and its relation to other metrics of current interest. Moreover, we also prove that another function, studied by O. Dovgoshey, P. Hariri, and M. Vuorinen is a metric and give upper and lower bounds for it.