Metric complexity is a Bryant--Tupper diversity
This work provides a theoretical link between two mathematical concepts, offering a new perspective on diversity measures in metric spaces, but it is incremental as it builds on existing frameworks without broad practical applications.
The paper connects metric complexity, an isometry-invariant of compact metric spaces, to Bryant--Tupper diversity, showing that metric complexity naturally produces an example of this diversity. It also demonstrates that this diversity is Minkowski-superadditive for compact subsets of the real line, unlike other examples in the literature.
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.