  A joint mix is a random vector with a constant component-wise sum. The
dependence structure of a joint mix minimizes some common objectives such as
the variance of the component-wise sum, and it is regarded as a concept of
extremal negative dependence. In this paper, we explore the connection between
the joint mix structure and popular notions of negative dependence in
statistics, such as negative correlation dependence, negative orthant
dependence and negative association. A joint mix is not always negatively
dependent in any of the above senses, but some natural classes of joint mixes
are. We derive various necessary and sufficient conditions for a joint mix to
be negatively dependent, and study the compatibility of these notions. For
identical marginal distributions, we show that a negatively dependent joint mix
solves a multi-marginal optimal transport problem for quadratic cost under a
novel setting of uncertainty. Analysis of this optimal transport problem with
heterogeneous marginals reveals a trade-off between negative dependence and the
joint mix structure.
