Diversity (mathematics)

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

For all finite ${\displaystyle A\subseteq \mathbb {R} ^{n}}$ if we define ${\displaystyle \delta (A)=\sum _{i}\max _{a,b}\left\{\left|a_{i}-b_{i}\right|\colon a,b\in A\right\}}$ then ${\displaystyle (\mathbb {R} ^{n},\delta )}$ is a diversity.

If T is a phylogenetic tree with taxon set X. For each finite ${\displaystyle A\subseteq X}$, define ${\displaystyle \delta (A)}$ as the length of the smallest subtree of T connecting taxa in A. Then ${\displaystyle (X,\delta )}$ is a (phylogenetic) diversity.

Let ${\displaystyle (X,d)}$ be a metric space. For each finite ${\displaystyle A\subseteq X}$, let ${\displaystyle \delta (A)}$ denote the minimum length of a Steiner tree within X connecting elements in A. Then ${\displaystyle (X,\delta )}$ is a diversity.