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.