Diversity (mathematics)

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

Given a set $X$, let $\wp_\mbox(X)$ be the set of finite subsets of $X$.
A diversity is a pair $(X,\delta)$ consisting of a set $X$ and a function $\delta \colon \wp_\mbox(X) \to \mathbb$ satisfying
* (D1) $\delta(A)\geq 0$, with $\delta(A)=0$ if and only if $\left, A\\leq 1$; and
* (D2) if $B\neq\emptyset$ 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 $\delta(A)=\max_ d(a,b)=\operatorname(A)$ for all $A\in\wp_\mbox(X)$ defines a diversity.

$L_1$ diversity

For all finite $A\subseteq\mathbb^n$ if we define $\delta(A)=\sum_i\max_\left\$ then $(\mathbb^n,\delta)$ 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 $\delta(A)$ denote
the minimum length of a Steiner tree within ''X'' connecting elements in ''A''. Then $(X,\delta)$ is a
diversity.

Truncated diversity

Let $(X,\delta)$ be a diversity. For all $A\in\wp_\mbox(X)$ define
$\delta^(A) = \max\left\$. Then if $k\geq 2$, $(X,\delta^)$ 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.

References

{{Reflist

Metric spaces