Suppose we have a function \(f(x)\) that is convex, which is math talk for it always curves upwards, meaning \(d^{2} f(x) / d x^{2}\) is positive. \[ \begin{align*} d g(y) &=y d x+x d y-d f(x) \\[4pt]...Suppose we have a function \(f(x)\) that is convex, which is math talk for it always curves upwards, meaning \(d^{2} f(x) / d x^{2}\) is positive. \[ \begin{align*} d g(y) &=y d x+x d y-d f(x) \\[4pt] &=y d x+x d y-y d x \\[4pt] &=x d y \end{align*}\] \end{equation}\), it’s clear that a second application of the Legendre transformation would get you back to the original \(\begin{equation}
