Legendre transformation

Summary

The Legendre transformation turns a strictly convex function $f$ into a new function $f^*$ of the slope $p = f'(x)$. It records the same information as $f$, but with slopes as the variable instead of points. Applied twice it gives back $f$, and the derivatives of $f$ and $f^*$ are inverse functions of each other. It is the classical case of the convex conjugate and is used in mechanics, thermodynamics and optimization; the definition gives the precise formula.