Knowlpedia

Biconjugate

The conjugate of the conjugate, which produces a canonical closed convex minorant of a function.

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

f∗∗  =  (f∗)∗,f^{**} \;=\; (f^*)^*,

where f∗f^* is the of ff.

The biconjugate f∗∗f^{**} is always a , and it satisfies f∗∗≤ff^{**}\le f pointwise. The key characterization is given by the : under standard hypotheses (e.g. ff ), one has f=f∗∗f=f^{**} exactly when ff is closed and convex.

Examples
  • If ff is closed and convex (for instance a norm or a quadratic), then f∗∗=ff^{**}=f.
  • If f=δCf=\delta_C is the indicator of a set C⊆RnC\subseteq\mathbb{R}^n, then δC∗∗=δconv‾(C)\delta_C^{**}=\delta_{\overline{\mathrm{conv}}(C)}, the indicator of the closed convex hull of CC (e.g. for C={−1,1}⊂RC=\{-1,1\}\subset\mathbb{R}, one gets δC∗∗=δ[−1,1]\delta_C^{**}=\delta_{[-1,1]}).