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)

Note that technically we still need to bridge the GF convexity to what we call strictly convex. Probably Real Function with Strictly Positive Second Derivative is Strictly Convex will do, but please do check. Otherwise, the above formulation can be said to match what GF tried to say. BTW, the very next page on GF has the definition using $\max$ or, in other words, $\sup$, so with appropriate care both notions are supported by GF. Still, I would recommend looking for a better source.--Julius (talk) 08:34, 6 December 2022 (UTC)