  Much of the existing work on the broadcast channel focuses only on the
sending of private messages. In this work we examine the scenario where the
sender also wishes to transmit common messages to subsets of receivers. For an
L user broadcast channel there are 2L - 1 subsets of receivers and
correspondingly 2L - 1 independent messages. The set of achievable rates for
this channel is a 2L - 1 dimensional region. There are fundamental constraints
on the geometry of this region. For example, observe that if the transmitter is
able to simultaneously send L rate-one private messages, error-free to all
receivers, then by sending the same information in each message, it must be
able to send a single rate-one common message, error-free to all receivers.
This swapping of private and common messages illustrates that for any broadcast
channel, the inclusion of a point R* in the achievable rate region implies the
achievability of a set of other points that are not merely component-wise less
than R*. We formerly define this set and characterize it for L = 2 and L = 3.
Whereas for L = 2 all the points in the set arise only from operations relating
to swapping private and common messages, for L = 3 a form of network coding is
required.
