---
tag: B1X92F7SJ6
kind: definition
name: Fenchel conjugate
msc: [52A41, 26B25, 90C46]
defines: T7V2VPY61S
uses: [03J5RAKB30, 07JBSZE0V4, 0R3H5PANRQ, 0GSZ7B0CGR]
sources:
  - url: https://en.wikipedia.org/wiki/Convex_conjugate
---

## Definition

Let $X$ be a real [[normed space|03J5RAKB30]] with continuous [[dual space|07JBSZE0V4]] $X^*$, and write $\ip{y}{x} = y(x)$ for $y \in X^*$ and $x \in X$.
Let $f \colon X \to [-\infty, +\infty]$ be a function.
The *Fenchel conjugate* (or *convex conjugate*) of $f$ is the function $f^* \colon X^* \to [-\infty, +\infty]$ given by
$$f^*(y) = \sup_{x \in X} \big(\ip{y}{x} - f(x)\big),$$
where $\sup$ is the [[supremum|0R3H5PANRQ]] in the [[extended real line|0GSZ7B0CGR]].
For $X = \R^n$, one identifies $X^*$ with $\R^n$ through the dot product.

## 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|ZQYMWVT1TT]] 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.