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.