Legendre transform

A Legendre transform of a differentiable strictly convex function $f:U\to\mathbb{R}$ on an open convex set $U\subseteq\mathbb{R}^n$ is the function $f^{\mathcal L}$ defined on the set $\nabla f(U)\subseteq\mathbb{R}^n$ by

Strict convexity ensures that for each $p\in\nabla f(U)$ there is a unique $x\in U$ with $p=\nabla f(x)$, so the value is well-defined.

The Legendre transform is the “attained” version of the Fenchel conjugate when $f$ is smooth enough that the supremum defining $f^*$ is achieved at the unique point with $p=\nabla f(x)$. It connects convex analysis with the derivative and differentiable map through the gradient map $x\mapsto\nabla f(x)$.

Examples:

• If $f(x)=\tfrac12\|x\|_2^2$ on $\mathbb{R}^n$, then $\nabla f(x)=x$ and $f^{\mathcal L}(p)=\tfrac12\|p\|_2^2$.
• If $f(x)=e^x$ on $\mathbb{R}$, then $\nabla f(x)=e^x$ so $p>0$ and $f^{\mathcal L}(p)=p\log p - p$ on $(0,\infty)$.