Trees, Tight-Spans and Point Configuration

TL;DR

This paper demonstrates that tight-spans of various metrics and maps can be uniformly represented using point configurations, enabling systematic analysis and extension of known results.

Contribution

It introduces a unified framework using point configurations to study tight-spans of diverse maps, including directed metrics and diversities, and extends existing results.

Findings

Unified representation of tight-spans via point configurations

Recovery of one-dimensional tight-span results for various maps

Extension of tight-span results to more general maps such as symmetric and asymmetric

Abstract

Tight-spans of metrics were first introduced by Isbell in 1964 and rediscovered and studied by others, most notably by Dress, who gave them this name. Subsequently, it was found that tight-spans could 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 extend these and other results to more general maps such as symmetric and unsymmetric maps.