A code \(\mathcal{C} \subset 2^{[n]}\) is open convex if there is a collection \(\mathcal{U}\) of open convex subsets of \(\mathbb{R}^d\) (for some \(d \geq 1\)) so that \(\mathrm{code}(\mathcal{U}) = \mathcal{C}\). If \(\mathcal{U}\) is instead a collection of closed convex subsets of \(\mathbb{R}^d\) and \(\mathrm{code}(\mathcal{U}) = \mathcal{C}\), then \(\mathcal{C}\) is closed convex.

It's easy to see (but technical to prove) that every open convex code is also closed convex (shrink each of the open sets by a small enough \(\varepsilon\) so that the intersection pattern is not altered, then take closure of the new sets), but surprisingly there are closed convex codes which are not open convex. An example of such a code is provided in Theorem 3.1 of Obstructions to convexity in neural codes by Lienkaemper, Shiu, and Woodstock. (In that paper, the authors use the term convex to convey what I have defined above as open convex.) There, they show that the code \[\mathcal{C} = \{2345, 123, 134, 145, 13, 14, 23, 34, 45, 3, 4,\varnothing\}\] is not open convex. However, there is a collection \(\mathcal{U}\) of closed convex subsets of \(\mathbb{R}^2\) for which \(\mathrm{code}(\mathcal{U}) = \mathcal{C}\), and one such collection is provided in a forthcoming paper by Giusti, Itskov, and myself.