Solution
= Solution
For the <convex conjugate>, use a dual scalar $q$ to avoid confusing it with the fixed centre $z$. Differentiating $qx-\lambda(x-z)^2/2$ gives $q-\lambda(x-z)=0$, so the unique maximizing point is $x=z+q/\lambda$. Substitution yields
$$
\boxed{E^*(q)=qz+\frac{q^2}{2\lambda}.}
$$
The positivity of $\lambda$ makes the maximized quadratic strictly concave and the <convex conjugate> finite for every real $q$.