A Monotonicity Principle for Information Theory

We establish a monotonicity principle for convex functions that enables high-level reasoning about capacity in information theory. Despite its simplicity, this single idea is remarkably applicable. It leads to a significant extension of algebraic information theory, a solution of the capacity reduction problem, intuitive graphical methods for comparing channels, new inequalities that provide useful estimates on the information transmitting capability of a channel operating in an unknown environment, further explication of the fascinating relationship between capacity and Euclidean distance, and the solution of an open problem in quantum steganography.