Diversity (mathematics) explained

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

X , let \wp _{fin(X)} be the set of finite subsets of X .A diversity is a pair (X,\delta) consisting of a set X and a function \delta \colon \wp _{fin(X)} \to R satisfying

(D1)

\delta(A)\geq 0 , with \delta(A) =0 if and only if \left|A\right|\leq 1

and

(D2) if

B ≠ \emptyset then \delta(A\cup C)\leq\delta(A\cup B) + \delta(B \cup C) .

Bryant and Tupper observe that these axioms imply monotonicity; that is, if

A\subseteq B , then \delta(A)\leq\delta(B) . 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

Let

(X,d) be a metric space. Setting \delta(A) =max _{a,b\in} d(a,b) = \operatorname{diam}(A) for all A\in\wp _{fin(X)} defines a diversity.L _{1} diversity

For all finite

A\subseteq R ^{n} if we define \delta(A) = \sum _{imax} _{a,b} \left\{\left| a _{i-b} _{i\right|\colon} a,b\in A\right\} then ( R ^{n,\delta)} is a diversity.

Phylogenetic diversity

If T is a phylogenetic tree with taxon set X. For each finite

A\subseteq X , define\delta(A) as the length of the smallest subtree of T connecting taxa in A. Then (X, \delta) is a (phylogenetic) diversity.

Steiner diversity

Let

(X, d) be a metric space. For each finite A\subseteq X , let \delta(A) denotethe minimum length of a Steiner tree within X connecting elements in A. Then (X,\delta) is adiversity.

Truncated diversity

Let

(X,\delta) be a diversity. For all A\in\wp _{fin(X)} define\delta ^{(k)} (A) = max \left\{\delta(B)\colon |B|\leq k, B\subseteq A\right\} . Then if k\geq 2 , (X,\delta ^{(k)} ) is a diversity.

Clique diversity

If

(X,E) is a graph, and \delta(A) is defined for any finite A as the largest clique of A, then (X,\delta) is a diversity.

Notes and References

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