割线引理，以及我为什么总在重推它The secant-slope lemma, and why I keep re-deriving it

Convexity arguments in optimisation almost always reduce to the same three inequalities about secant slopes. I have derived them from scratch at least four times, which is three times too many, so the derivation lives here now.

Statement

Let $f$ be convex on an interval $I$, and let $x \lt y \lt z$ be points of $I$. Then

The middle term is what makes this useful: it says the average slope over the whole interval lies between the two partial average slopes.

Proof

Write $y$ as a convex combination of the endpoints. Since $x \lt y \lt z$,

By convexity of $f$:

Subtract $f(x)$ from both sides and use $t\,f(z) - t\,f(x) = t\,(f(z) - f(x))$:

Dividing by $y - x = t\,(z - x)$, which is strictly positive, gives the left inequality of Lemma 1:

For the right inequality of Lemma 1, multiply out the target inequality by the positive quantity $(z - x)(z - y)$ and collect terms; what is left is

which is convexity again, with the weights summing to one. So the two inequalities are the same statement read in opposite directions, and no separate argument is needed. $\square$

Why it is worth writing down

The lemma is trivial. Its value is that it converts "by convexity" — which is where hand-waving usually starts — into a specific inequality you can point at. Most of the arguments I have watched fall apart did so because a step assumed this lemma without stating it, and then applied it to a function that was not convex.