For a function on a real vector space, its convex conjugate isWhen is differentiable and strictly convex, the maximizing point satisfies .
For a proper convex function and its convex conjugate ,Equality holds exactly when , or equivalently .
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The convex conjugate, also known as the Legendre-Fenchel transform, is a concept in convex analysis and optimization that is used to transform a convex function into another function.