# 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.