  Marton's inner bound is the best known achievable region for a general
discrete memoryless broadcast channel. To compute Marton's inner bound one has
to solve an optimization problem over a set of joint distributions on the input
and auxiliary random variables. The optimizers turn out to be structured in
many cases. Finding properties of optimizers not only results in efficient
evaluation of the region, but it may also help one to prove factorization of
Marton's inner bound (and thus its optimality). The first part of this paper
formulates this factorization approach explicitly and states some conjectures
and results along this line. The second part of this paper focuses primarily on
the structure of the optimizers. This section is inspired by a new binary
inequality that recently resulted in a very simple characterization of the
sum-rate of Marton's inner bound for binary input broadcast channels. This
prompted us to investigate whether this inequality can be extended to larger
cardinality input alphabets. We show that several of the results for the binary
input case do carry over for higher cardinality alphabets and we present a
collection of results that help restrict the search space of probability
distributions to evaluate the boundary of Marton's inner bound in the general
case. We also prove a new inequality for the binary skew-symmetric broadcast
channel that yields a very simple characterization of the entire Marton inner
bound for this channel.
