Fenchel's duality theorem

In mathematics, Fenchel's duality theorem is a result in the theory of convex functions named after Werner Fenchel.

Let ƒ be a proper convex function on Rⁿ and let g be a proper concave function on Rⁿ. Then, if regularity conditions are satisfied,

where ƒ ^* is the convex conjugate of ƒ (also referred to as the Fenchel–Legendre transform) and g _* is the concave conjugate of g. That is,

Let X and Y be Banach spaces, ${\displaystyle f:X\to \mathbb {R} \cup \{+\infty \}}$ and ${\displaystyle g:Y\to \mathbb {R} \cup \{+\infty \}}$ be convex functions and ${\displaystyle A:X\to Y}$ be a bounded linear map. Then the Fenchel problems:

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