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
fin
be the set of finite subsets of
A diversity is a pair
consisting of a set
satisfying
δ (
δ (
Bryant and Tupper observe that these axioms imply monotonicity; that is, if
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: Let
be a metric space.
defines a diversity.
For all finite
if we define
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.
is a (phylogenetic) diversity.
be a metric space.
For each finite
denote the minimum length of a Steiner tree within X connecting elements in A.
define
is defined for any finite A as the largest clique of A, then
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