Biconjugate

A biconjugate of an extended-real-valued function $f:\mathbb{R}^n\to(-\infty,+\infty]$ is the function

where $f^*$ is the Fenchel conjugate of $f$.

The biconjugate $f^{**}$ is always a closed convex function , and it satisfies $f^{**}\le f$ pointwise. The key characterization is given by the Fenchel–Moreau theorem : under standard hypotheses (e.g. $f$ proper ), one has $f=f^{**}$ exactly when $f$ is closed and convex.

Examples:

• If $f$ is closed and convex (for instance a norm or a quadratic), then $f^{**}=f$.
• If $f=\delta_C$ is the indicator of a set $C\subseteq\mathbb{R}^n$, then $\delta_C^{**}=\delta_{\overline{\mathrm{conv}}(C)}$, the indicator of the closed convex hull of $C$ (e.g. for $C=\{-1,1\}\subset\mathbb{R}$, one gets $\delta_C^{**}=\delta_{[-1,1]}$).