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Diversity (mathematics)

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In mathematics, a diversity is a generalization of the concept of metric space. The concept was introduced in 2012 by Bryant and Tupper,[1] who call diversities "a form of multi-way metric".[2] The concept finds application in nonlinear analysis.[3]

Given a set , let be the set of finite subsets of . A diversity is a pair consisting of a set and a function satisfying

(D1) , with if and only if

and

(D2) if then .

Bryant and Tupper observe that these axioms imply monotonicity; that is, if , then . They state that the term "diversity" comes from the appearance of a special case of their definition in work on phylogenetic and ecological diversities. They give the following examples:

Diameter diversity

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Let be a metric space. Setting for all defines a diversity.

L1 diversity

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For all finite if we define then is a diversity.

Phylogenetic diversity

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If T is a phylogenetic tree with taxon set X. For each finite , define as the length of the smallest subtree of T connecting taxa in A. Then is a (phylogenetic) diversity.

Steiner diversity

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Let be a metric space. For each finite , let denote the minimum length of a Steiner tree within X connecting elements in A. Then is a diversity.

Truncated diversity

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Let be a diversity. For all define . Then if , is a diversity.

Clique diversity

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If is a graph, and is defined for any finite A as the largest clique of A, then is a diversity.

References

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  1. ^ Bryant, David; Tupper, Paul (2012). "Hyperconvexity and tight-span theory for diversities". Advances in Mathematics. 231 (6): 3172–3198. arXiv:1006.1095. doi:10.1016/j.aim.2012.08.008.
  2. ^ Bryant, David; Tupper, Paul (2014). "Diversities and the geometry of hypergraphs". Discrete Mathematics and Theoretical Computer Science. 16 (2): 1–20. arXiv:1312.5408.
  3. ^ Espínola, Rafa; Pia̧tek, Bożena (2014). "Diversities, hyperconvexity, and fixed points". Nonlinear Analysis. 95: 229–245. doi:10.1016/j.na.2013.09.005. hdl:11441/43016. S2CID 119167622.