Hyperconvexity and Tight Span Theory for Diversities

TL;DR

This paper generalizes the concepts of tight spans and hyperconvexity from metric spaces to diversities, expanding the theoretical framework and potential applications in data analysis.

Contribution

It introduces the notion of diversity tight spans and hyperconvexity, extending existing metric space theories to a broader class of diversities.

Findings

Diversity tight spans can be constructed similarly to metric tight spans.

Hyperconvexity properties extend naturally to diversities.

The generalized theory has potential applications in data visualization and classification.

Abstract

The tight span, or injective envelope, is an elegant and useful construction that takes a metric space and returns the smallest hyperconvex space into which it can be embedded. The concept has stimulated a large body of theory and has applications to metric classification and data visualisation. Here we introduce a generalisation of metrics, called diversities, and demonstrate that the rich theory associated to metric tight spans and hyperconvexity extends to a seemingly richer theory of diversity tight spans and hyperconvexity.