What is the relation between two different functions, say $g(x)$ and $f(x)$, which have the same Legendre–Fenchel transformation $h(s)$?

\begin{equation} h(s) = \sup_{x\in I}\{sx - f(x)\} \quad \quad x \in I \end{equation} \begin{equation} h(s) = \sup_{x\in I'}\{sx - g(x)\} \quad \quad x \in I^{'} \end{equation} Must they be the same? What are the conditions for $g(x) = f(x)$?


Legendre transformations preserve information. Indeed if it is possible to recreate $y$ from $y^{*}$, then no information can have gotten lost. This is the case under the assumption of convexity of $y$. I refer to this paper.

Observe, though, that this does only hold if the function we started with was convex or concave. Since the Legendre transform still makes sense if we have local deviations from convexity or concavity (say, an overall convex function with a local concave “bump”), we might ask what now happens after two Legendre transforms. The answer is that we recover the convex (concave) envelope of the original function. This finding plays an important role in the theory of phase transitions.

For sufficient and necessary conditions, as Giuseppe Negro was suggesting, we have that $f=f^{{**}}$ (biconjugate) if and only if $f$ is convex and lower semi-continuous (under some assumptions on the domain), by the Fenchel–Moreau theorem.


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