Legendre Transform is Involution

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Theorem

The Legendre transform is an Involution.


Proof

Let $\map f x$ be a strictly convex real function.

Let $p = \map {f'} x$.

By definition of the Legendre transform, the transformed real function is of the form:

$\map {f^*} p = - \map f {\map x p} + p \map x p$

By Legendre Transform of Strictly Convex Real Function is Strictly Convex, $f^*$ is strictly convex.


Let $t = \map { {f^*}'} p$.

Let $\map {\paren {f^*}^*} t = - \map {f^*} {\map p t} + t \map p t$.

Then:

\(\displaystyle \paren {f^*}^*\) \(=\) \(\displaystyle -f^* + t p\)
\(\displaystyle \) \(=\) \(\displaystyle -\paren {-f + p t} + t p\)
\(\displaystyle \) \(=\) \(\displaystyle f\)

$t$ is an arbitrary independent variable, hence can be renamed.

Set $t = x$.

Then:

$\tuple {t,f^{**} } = \tuple {x, f}$

which is the original pair of function and its variable.

Hence, by definition, the Legendre transform is an involution.

$\blacksquare$


Sources