Affine covariance of the convex conjugate ($g^{\ast }(p)=\lambda f^{\ast }(p/\lambda )+p^T{x}_0+\mu$)

If $g(x)=\lambda f(x-x_0)-\mu$ with $\lambda >0$, substitution into the supremum defining the convex conjugate givesThe positivity of $\lambda$ allows it to pass through the supremum without reversing it. Negative scaling need not obey this rule for the convex conjugate, even when a formal stationary-value transform does.