Convex conjugate In mathematics , convex conjugation is a generalization of the Legendre transformation . It is also known as Legendre-Fenchel transformation or Fenchel transformation . Definition Let X be a real normed vector space , and let X^ { * } be the dual space to X . Denote the dual pairing by \langle \cdot , \cdot \rangle : X^ { * } \times X \to \mathbb { R } For a function f : X \to \mathbb { R } \cup \ { + \infty \ } taking values on the extended real number line the convex conjugate f^\star : X^ { * } \to \mathbb { R } \cup \ { + \infty \ } is defined by f^ { \star } \left ( x^ { * } \right ) : = \sup \left \ { \left. \left\langle x^ { * } , x \right\rangle - f \left ( x \right ) \right| x \in X \right\ } , or , equivalently , by f^ { \star } \left ( x^ { * } \right ) : = - \inf \left \ { \left. f \left ( x \right ) - \left\langle x^ { * } , x \right\rangle \right| x \in X \right\ } . Examples The convex conjugate of an affine function f ( x ) = \left\langle a , x \right\rangle - b , \ , a \in \mathbb { R } ^n , b \in \mathbb { R } is f^\star\left ( x^ { * } \right ) = \begin { cases } b , & x^ { * } = a \\ \infty , & x^ { * } \ne a \end { cases } The convex conjugate of the absolute value function f ( x ) = \left| x \right| is f^\star\left ( x^ { * } \right ) = \begin { cases } 0 , & \left|x^ { * } \right| \le 1 \\ \infty , & \left|x^ { * } \right| > 1 \end { cases } The convex conjugate of the exponential function is \exp^\star\left ( x^ { * } \right ) = \begin { cases } x^ { * } \ln x^ { * } - x^ { * } , & x^ { * } > 0 \\ 0 , & x^ { * } = 0 \\ \infty , & x^ { * } < 0 \end { cases } Convex conjugate and Legendre transform of the exponential function agree except that the domain of the convex conjugate is strictly larger as the Legendre transform is only defined for positive real numbers . Properties The convex conjugate of a closed convex function is again a closed convex function . The convex conjugate of a polyhedral convex function ( a convex function with polyhedral epigraph ) is again a polyhedral convex function . Convex-conjugation is order-reversing : if f   ? g then f *   ? g * . Here f   ? g if and only if f ( x )   ? g ( x ) for all x . Biconjugate The convex conjugate of a function is always lower semi-continuous . The biconjugate f ** ( the convex conjugate of the convex conjugate ) is also the closed convex hull , i.e. the largest lower semi-continuous convex function smaller than f . Therefore , f = f ** if and only if f is convex and lower semi-continuous . Fenchel 's inequality For any proper convex function f and its convex conjugate f * Fenchel 's inequality ( also known as the Fenchel-Young inequality ) holds : \left\langle p , x \right\rangle \le f ( x ) + f^\star ( p ) Behavior under linear transformations Let A be a linear transformation from R n to R m . For any convex function f on R n , one has \left ( A f\right ) ^\star = f^\star A^\star where A * is the adjoint operator of A defined by \left \langle Ax , y^\star \right \rangle = \left \langle x , A^\star y^\star \right \rangle A closed convex function f is symmetric with respect to a given set G of orthogonal linear transformations , f\left ( A x\right ) = f ( x ) , \ ; \forall x , \ ; \forall A \in G if and only if its convex conjugate f * is symmetric with respect to G . Infimal convolution The infimal convolution of two functions f and g is defined as \left ( f \star_\inf g\right ) ( x ) = \inf \left \ { f ( x-y ) + g ( y ) \ , | \ , y \in \mathbb { R } ^n \right \ } Let f 1 , … , f m be proper convex functions on R n . Then \left ( f_1 \star_\inf \cdots \star_\inf f_m \right ) ^\star = f_1^\star + \cdots + f_m^\star References Arnol'd , Vladimir Igorevich ( 1989 ) . Mathematical Methods of Classical Mechanics ( second edition ) . Springer . ISBN 0-387-96890-3 . Rockafellar , Ralph Tyrell ( 1996 ) . Convex Analysis . Princeton University Press . ISBN 0-691-01586-4 . Categories : Convex analysis | Duality theories | Transforms In other languages : Español Arnol'd , Vladimir Igorevich Mathematical Methods of Classical Mechanics ( second edition ) Springer 1989 ISBN 0-387-96890-3 Rockafellar , Ralph Tyrell Convex Analysis Princeton University Press 1996 ISBN 0-691-01586-4 