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 , 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

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

For all finite  if we define  then  is a diversity. Phylogenetic diversity

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

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

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

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

Metric spaces