We study certain inner bounds in broadcast channel that use auxiliary random
variables and reformulate it in terms of concave envelopes of some functions
on
input distributions. This reformulation then induces  new types of
"factorization problems" whose establishment would then yield the optimality
of
the inner bounds. This could potentially lead to a new way of establishing
outer
bounds.

To provide preliminary justification for the introduction of this new
formulation, we show how certain old non-trivial results on extremal
distributions can be shown using very elementary techniques. This work is an
attempt to capture the key intuition behind some of the recent developments in
the broadcast channel.