Convexity & Jensen

Balance minimises what curves upward.

A convex function's chords lie above its graph; Jensen's inequality is that picture averaged over $n$ points: the average of the values is at least the value at the average, with equality when all inputs coincide. Concave functions run the same statement downhill — so a fixed sum of inputs pins a minimum for convex targets and a maximum for concave ones, and the direction is read off the curvature.

Most named inequalities are Jensen with a chosen function: $\ln x$ gives AM–GM, $1/x$ gives AM–HM, $x^2$ gives QM–AM. Learning the parent theorem replaces the list.

When Jensen's shape is missing, the tangent line trick recovers it: guess the equality point, write the supporting line there, prove the one-variable inequality, and sum. It is Jensen's own proof, deployed by hand where the packaged theorem does not reach.

Where it appears on Lemma: Level 7 (convexity & Jensen's inequality).

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