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Diversities are a generalization of metric spaces in which a non-negative value is assigned to all finite subsets of a set, rather than just to pairs of points. Here we provide an analogue of the theory of negative-type metrics for diversities. We introduce negative-type diversities, and show that, as in the metric space case, they are a generalization of \(L_1\)-embeddable diversities. We provide a number of characterizations of negative-type diversities, including a geometric characterization. Much of the recent interest in negative-type metrics stems from the connections between metric embeddings and approximation algorithms. We extend some of this work into the diversity setting, showing that lower bounds for embeddings of negative-type metrics into \(L_1\) can be extended to diversities by using recently established extremal results on hypergraphs.

Negative-Type Diversities, a Multi-dimensional Analogue of Negative-Type Metrics