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?
2026-04-24 03:41:22.1777002082
Are convex sets solid?
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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.