# Uniqueness of the Legendre-Fenchel Transformation

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

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