Legendre–Fenchel transform

A Legendre–Fenchel transform of an extended-real-valued function $f:\mathbb{R}^n\to(-\infty,+\infty]$ is the function $f^*:\mathbb{R}^n\to[-\infty,+\infty]$ defined by

f^*(y) \;=\; \sup_{x\in\mathbb{R}^n}\big(\langle y,x\rangle - f(x)\big).

where $\langle y,x\rangle=\sum_{i=1}^n y_i x_i$ is the Euclidean inner product.

Properties and limiting cases

If $f\equiv+\infty$ , then $f^*\equiv-\infty$ ; otherwise $f$ is finite somewhere and $f^*$ never takes the value $-\infty$ .

This is exactly the Fenchel conjugate; the “Legendre” terminology is common even when $f$ is not differentiable, and it underlies Fenchel–Young inequality and biconjugation. When $f$ is smooth and strictly convex, the transform agrees with the classical Legendre transform on the range of $\nabla f$ .

Examples

If $f(x)=|x|$ on $\mathbb{R}$ , then $f^*(y)=0$ for $|y|\le 1$ and $f^*(y)=+\infty$ for $|y|>1$ .
If $f=\delta_C$ is the indicator of a set $C$ , then $f^*(y)=\sup_{x\in C}\langle y,x\rangle$ (the support function of $C$ ).