If $X'$ is the topological dual of a Banach space, then is that true that a convex set is closed (for the norm on $X'$ given by $\lVert f \rVert_{X'} := \sup \frac{\langle f , x \rangle}{\lVert x \rVert_{X}}$) if and only if it is weak*-closed?

If it is false, is there a class of infinite-dimensional spaces on which it is true?

(I know that a closed set is weakly closed if and only if it is closed but I could not find such a result (or its negation) for the weak*-topology.)

If $X'$ is the topological dual of a Banach space, then is that true that a convex set is closed (for the norm on $X'$ given by $\lVert f \rVert_{X'} := \sup \frac{\langle f , x \rangle}{\lVert x \rVert_{X}}$) if and only if it is weak*-closed?

If it is false, is there a class of infinite-dimensional spaces on which it is true?

(I know that a convex set is weakly closed if and only if it is closed but I could not find such a result (or its negation) for the weak*-topology.)