Let $X$ be a closed set in a Euclidean space of finite dimension and suppose that its convex hull $H$ is open. I can prove that, in this case, $H$ is a Cartesian product of a line with an open convex set of dimension lower by one. (See https://mathstodon.xyz/@11011110/103381278180283137 for proof sketch.) It's also not hard to realize any such product as an open convex hull of a closed set. Does this fact appear in the literature anywhere?