Fenchel conjugate

A Fenchel conjugate 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

where $\langle y,x\rangle$ denotes the Euclidean pairing.

This operation (also called the Legendre–Fenchel transform ) turns $f$ into a (possibly extended-real) convex function , and it is the basic object behind Fenchel–Young inequality and convex duality . The definition uses the supremum over all affine functionals $x\mapsto \langle y,x\rangle - f(x)$.

Examples:

• If $f(x)=\tfrac12\|x\|_2^2$ on $\mathbb{R}^n$, then $f^*(y)=\tfrac12\|y\|_2^2$.
• If $f=\delta_C$ is the indicator of a nonempty set $C$ (i.e. $\delta_C(x)=0$ for $x\in C$ and $+\infty$ otherwise), then $f^*(y)=\sup_{x\in C}\langle y,x\rangle$, the support function of $C$.