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

Geometry of maximum-entropy proofs: stationary points, convexity, Legendre transforms, exponential families