On open and closed convex codes

Combinatorial codes, i.e. subsets of the power set, naturally arise as the language of neural networks in the brain and silicon. A number of combinatorial codes in the brain arise as a pattern of intersections of convex sets in a Euclidean space. Such codes are called convex. Not every code is convex. What makes a code convex and what determines its embedding dimension is still poorly understood. Here we establish that the codes that arise from open convex sets and the codes that arise from closed convex sets are distinct classes of codes. We prove that open convexity is inherited from a sub-code with the same simplicial complex. Finally we prove that codes that contain all intersections of its maximal codewords are both open and closed convex, and provide an upper bound for their embedding dimension.