Legendre transforms appear all over physics. For instance, in statistical mechanics, they allow us to move between descriptions in terms of different thermodynamic potentials. Similarly, in quantum field theory, they are used to construct the effective action $\Gamma[\varphi]$ (the generating functional of one-particle irreducible correlators) from $W[J]$, the generating functional of connected correlators.
The thing is, you often see these transformations in two different forms. It might be either $$\Gamma[\varphi] = \sup_J(J \cdot \varphi - W[J]),$$ see e.g. here, or just $$\Gamma[\varphi] = J \cdot \varphi - W[J].$$ as it is written here and on Wikipedia. I understand taking the supremum ensures that $\Gamma[\varphi]$ ends up a convex function of $\varphi$. So why do some people worry about this and others don't? Is there any more to it, perhaps with regard to invertibility?
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