Trees, tight-spans and point configurations

Tight-spans of metrics were first introduced by Isbell in 1964 and rediscovered and studied by others, most notably by Dress in 1984, who gave them this name. Subsequently, it has been found that tight-spans can be defined for more general maps, such as directed metrics and distances, and more recently for diversities. In this paper, we show that all of these tight-spans, as well as some related constructions, can be defined in terms of point configurations. This provides a useful way in which to study these objects in a unified and systematic way. We also show that by using point configurations we can recover results concerning one-dimensional tight-spans for all of the maps we consider, as well as extending these and other results to more general maps such as symmetric and asymmetric maps.