  We describe certain extremalities for Gallager's $E_0$ function evaluated
under the uniform input distribution for binary input discrete memoryless
channels. The results characterize the extremality of the $E_0(\rho)$ curves of
the binary erasure channel and the binary symmetric channel among all the
$E_0(\rho)$ curves that can be generated by the class of binary discrete
memoryless channels whose $E_0(\rho)$ curves pass through a given point
$(\rho_0, e_0)$, for some $\rho_0 > -1$.
