Lipschitz means and mixers on metric spaces

The standard arithmetic measures of center, the mean, and the median, have natural topological counterparts that have been widely used in continuum theory. In the context of metric spaces, it is natural to consider the Lipschitz continuous versions of the mean and median. We show that they are related to familiar concepts of the geometry of metric spaces: the bounded turning property, the existence of quasisymmetric parameterization, and others.