Are convex sets solid?

Question

In a normed vector space, call a set $X$ solid if whenever $x\in X$ and $B(x, r)\subseteq\overline X$, we have $B(x, \epsilon r)\subseteq X$ for some $0<\epsilon<1$.

Are convex sets solid?

Answer 1

Not necessarily. In an infinite-dimensional normed space, there are dense proper subspaces. As a subspace, these are convex, and as proper subspaces, they have empty interior. But their closure has nonempty interior.