In mathematics and mathematical optimization, the convex conjugate of a function is a generalization of the Legendre transformation which applies to non-convex functions. It is also known as Legendre–Fenchel transformation or Fenchel transformation (after Adrien-Marie Legendre and Werner Fenchel). It is used to transform an optimization problem into its corresponding dual problem, which can often be simpler to solve.
Let be a real topological vector space, and let be the dual space to . Denote the dual pairing by