The Least Degraded and the Least Upgraded Channel with respect to a Channel Family

Given a family of binary-input memoryless output-symmetric (BMS) channels having a fixed capacity, we derive the BMS channel having the highest (resp. lowest) capacity among all channels that are degraded (resp. upgraded) with respect to the whole family. We give an explicit characterization of this channel as well as an explicit formula for the capacity of this channel.