Definition talk:Legendre Transform

There is considerably more detail in the source work than has been detailed here.

The variable $(1): \quad p = \map {f'} x$ is called the tangential coordinate, and this would be worth adding as a self-contained definition page in its own right.

Then it goes on to state (which will need a separate proof page here and needs to be brought forward to this page because it is crucial) that for a strictly convex function (same applies to strictly concave) any point on the curve $\eta = \map f x$ is uniquely determined by the slope of the tangent. Hence the new function $f^*$ (the notation $H$ is used in G&F) is defined as:
 * $(2): \quad \map H p = -\map f {\xi} + p \xi$

where $\xi$ is regarded as the function of $p$ obtained by solving $(1)$.

G&F use $\xi$ for both the independent variable of $f$ and the value of the function of $p$, which is confusing because now it is not clear what is what in $(2)$.

Can another source be consulted? --prime mover (talk) 07:02, 5 December 2022 (UTC)

The statement
The current statement is not understandable due to lack of formality.

How about the following? --Usagiop (talk) 10:25, 5 December 2022 (UTC)

Let $f : \R \to \R$ be a twice differentiable function.

Let $f$ be strictly convex, i.e.
 * $\forall x \in \R : \map {f' '} x > 0$

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

where $\map x p \in \R$ is the unique number such that:
 * $p = \map {f'} {\map x p}$

Looks good to me. If no one objects, feel free to implement.--Julius (talk) 19:15, 5 December 2022 (UTC)


 * Now I realized that this is still wrong. We need a differentiability assumption on $f$ to use the above statement as the definition. --Usagiop (talk) 20:39, 5 December 2022 (UTC)


 * The definition $\map {f^*} p := \sup_{x \in \R} \paren {x p - \map f x}$ works for any types of $f$. Then, the above statement is a theorem. Unfortunately, I have no good reference book. --Usagiop (talk) 20:43, 5 December 2022 (UTC)


 * Seriously, we prefer all definitions to be backed by a reference work. Uncorroborated knowledge is unreliable. --prime mover (talk) 20:47, 5 December 2022 (UTC)


 * I checked the reference (I.M. Gelfand and S.V. Fomin) and fond out that their convexity means $f' '>0$. Especially, the differentiability is assumed. I improved the above statement, correspondingly. It should be now OK. --Usagiop (talk) 00:00, 6 December 2022 (UTC)