Showing that set of all Hahn-Banach extensions of a functional is closed and has empty interior but need not be compact.

Question

Let $Y$ be a subspace of a normed linear space $X$ and $g\in Y'$. How do we show that the set of all Hahn-Banach extensions of $g$ to $X$ is a non-empty, convex, closed and bounded subset of $X'$ with empty interior but need not be compact.

1. The non-emptiness is just the Hahn-Banach theorem.
2. The boundedness follows from that all the extension has same norm as of $g$.
3. I can show that it is convex.

I am stuck with how to show it is closed, empty interior and need not be compact.