\begin{definition}[Definition:Legendre Transform]
Let $\map f x$ be a strictly convex real function.
Let $p = \map {f'} x$.
Let $\map {f^*} p = - \map f{\map x p} + p \map x p$.
{{explain|The above seems to imply that $x$ is a function.<br/>Yes it does, doesn't it. Exactly what it does mean is to be added to this page, and if necessary a new definition page is needed to specify it.}}
The '''Legendre Transform on $x$ and $f$''' is the mapping of the variable and function pair:
:$\paren{x, \map f x} \to \paren{p, \map {f^*} p}$
{{expand|generalise, add properties and connect with Young's inequality<br/>But not in here, do it somewhere else.}}
{{NamedforDef|Adrien-Marie Legendre|cat = Legendre}}
\end{definition}