What are the sets on which norm-closedness implies weakly closedness?

Let $ X$ be a Banach space. Let $ C$ be a convex, and normed-closed subset of $ X$ . It is well-known that $ C$ becomes weakly closed subset of $ X$ . I want to know is there any well-know class of non convex sets which has this property?

i.e., a class of sets in $ X$ , not necessary convex on which, norm-closed implies weakly-closed.