  This note is a geometric commentary on maximum-entropy proofs. Its purpose is
to illustrate the geometric structures involved in such proofs, to explain more
in detail why the maximization of the entropy can be turned into the
minimization of a potential function, and to show how Lagrange transforms
emerge from this. A synopsis of the main functions involved in the proof and of
their very different properties is given at the end, together with a brief
discussion of exponential families of probabilities, which also appear in the
proof.
