Let A : R → R be a linear transformation. For any convex function f on R, one has ( A f ) ⋆ ⋆ = f ⋆ ⋆ A ⋆ ⋆ where A * is the adjoint operator of A defined by ⟨ A x , y ⋆ ⋆ ⟩ = ⟨ x , A ⋆ ⋆ y ⋆ ⋆ ⟩ , and Af is the push-forward of f along A ( A f ) ( y ) = inf { f ( x ) : x ∈ ∈ X , A x = y } .