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

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 $C$).