  The key result of this paper is to find all the joint distributions of random
vectors whose sums $S=X_1+\ldots+X_d$ are minimal in convex order in the class
of symmetric Bernoulli distributions. The minimal convex sums distributions are
known to be strongly negatively dependent. Beyond their interest per se, these
results enable us to explore negative dependence within the class of copulas.
In fact, there are two classes of copulas that can be built from multivariate
symmetric Bernoulli distributions: the extremal mixture copulas, and the FGM
copulas. We study the extremal negative dependence structure of the copulas
corresponding to symmetric Bernoulli vectors with minimal convex sums and we
explicitly find a class of minimal dependence copulas. Our main results stem
from the geometric and algebraic representations of multivariate symmetric
Bernoulli distributions, which effectively encode several of their statistical
properties.
