  We explore the class of exchangeable Bernoulli distributions building on
their geometrical structure. Exchangeable Bernoulli probability mass functions
are points in a convex polytope and we have found analytical expressions for
their extremal generators. The geometrical structure turns out to be crucial to
simulate high dimensional and negatively correlated binary data. Furthermore,
for a wide class of statistical indices and measures of a probability mass
function we are able to find not only their sharp bounds in the class, but also
their distribution across the class. Estimate and testing are also addressed.
