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

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Abstract

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.

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Authors and Affiliations

  1. BGI-Tianjin, BGI-Shenzhen, Tianjin, 300308, China

    Pei Wu

  2. Department of Mathematics and Statistics, University of Otago, Dunedin, New Zealand

    David Bryant

  3. Department of Mathematics, Simon Fraser University, Burnaby, Canada

    Paul Tupper

Authors
  1. Pei Wu
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  2. David Bryant
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  3. Paul Tupper
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Correspondence to David Bryant.

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Wu, P., Bryant, D. & Tupper, P. Negative-Type Diversities, a Multi-dimensional Analogue of Negative-Type Metrics. J Geom Anal 31, 1703–1720 (2021). https://doi.org/10.1007/s12220-019-00321-0

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