Metric complexity is a Bryant--Tupper diversity

TL;DR

This paper explores the connection between metric complexity and diversity, showing that metric complexity naturally induces a diversity measure that is Minkowski-superadditive on the real line.

Contribution

It establishes a link between metric complexity and Bryant--Tupper diversity, demonstrating that metric complexity can generate a new diversity measure with specific properties.

Findings

Metric complexity induces a diversity measure.

The diversity from metric complexity is Minkowski-superadditive.

The connection provides new insights into metric-sensitive diversity.

Abstract

The metric complexity (sometimes called Leinster--Cobbold maximum diversity) of a compact metric space is a recently introduced isometry-invariant of compact metric spaces which generalizes the notion of cardinality, and can be thought of as a metric-sensitive analogue of maximum entropy. On the other hand, the notion of diversity introduced by Bryant and Tupper is an assignment of a real number to every finite subset of a fixed set, which generalizes the notion of a metric. We establish a connection between these concepts by showing that the former quantity naturally produces an example of the latter. Moreover, in contrast to several examples in the literature, the diversity that arises from metric complexity is Minkowski-superadditive for compact subsets of the real line.