Generalization of metric spaces
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
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