  Entropy, under a variety of names, has long been used as a measure of
diversity in ecology, as well as in genetics, economics and other fields. There
is a spectrum of viewpoints on diversity, indexed by a real parameter q giving
greater or lesser importance to rare species. Leinster and Cobbold proposed a
one-parameter family of diversity measures taking into account both this
variation and the varying similarities between species. Because of this latter
feature, diversity is not maximized by the uniform distribution on species. So
it is natural to ask: which distributions maximize diversity, and what is its
maximum value?
  In principle, both answers depend on q, but our main theorem is that neither
does. Thus, there is a single distribution that maximizes diversity from all
viewpoints simultaneously, and any list of species has an unambiguous maximum
diversity value. Furthermore, the maximizing distribution(s) can be computed in
finite time, and any distribution maximizing diversity from some particular
viewpoint q > 0 actually maximizes diversity for all q.
  Although we phrase our results in ecological terms, they apply very widely,
with applications in graph theory and metric geometry.
