Fenchel conjugate

Remarks

Example: $f(x) = \frac12 x^2$ on $\R$. For fixed $y$, the function $x \mapsto yx - \frac12 x^2$ is maximal at $x = y$, so $f^*(y) = \frac12 y^2$.

Example without differentiability: $f(x) = \abs{x}$ on $\R$. If $\abs{y} \le 1$ then $yx - \abs{x} \le 0$ with equality at $x = 0$, so $f^*(y) = 0$; if $\abs{y} > 1$ then $yx - \abs{x} \to +\infty$ along $x = t \sgn(y)$, so $f^*(y) = +\infty$. The classical formula of the Legendre transformation does not apply to this $f$, because $f$ is not differentiable at $0$.

Non-example of the identity $f^{**} = f$: for $f(x) = -x^2$ on $\R$, every $f^*(y)$ is $+\infty$, so $f^{**} \equiv -\infty \neq f$; this $f$ is not convex.

Every conjugate $f^*$ is convex and lower semicontinuous, as a supremum of affine continuous functions. The Fenchel–Young inequality $f(x) + f^*(y) \ge \ip{y}{x}$ holds whenever the left side is defined. The Fenchel–Moreau theorem states that $f^{**} = f$ on $X$ when $f$ is proper, convex and lower semicontinuous.