Editing Convex conjugate (section)

=== Behavior under linear transformations ===
Let <math>A : X \to Y</math> be a [[bounded linear operator]]. For any convex function <math>f</math> on <math>X,</math> 

:<math>\left(A f\right)^{*} = f^{*} A^{*}</math>

where

:<math>(A f)(y) = \inf\{ f(x) : x \in X , A x = y \}</math>

is the preimage of <math>f</math> with respect to <math>A</math> and <math>A^{*}</math> is the [[adjoint operator]] of <math>A.</math><ref>Ioffe, A.D. and Tichomirov, V.M. (1979), ''Theorie der Extremalaufgaben''. [[Deutscher Verlag der Wissenschaften]]. Satz 3.4.3</ref>

A closed convex function <math>f</math> is symmetric with respect to a given set <math>G</math> of [[orthogonal matrix|orthogonal linear transformation]]s,

:<math>f(A x) = f(x)</math> for all <math>x</math> and all <math>A \in G</math>

if and only if its convex conjugate <math>f^{*}</math> is symmetric with respect to <math>G.</math>