  Neural codes serve as a language for neurons in the brain. Convex codes,
which arise from the pattern of intersections of convex sets in Euclidean
space, are of particular relevance to neuroscience. Not every code is convex,
however, and the combinatorial properties of a code that determine its
convexity are still poorly understood. Here we find that a code that can be
realized by a collection of open convex sets may or may not be realizable by
closed convex sets, and vice versa, establishing that open convex and closed
convex codes are distinct classes. We also prove that max intersection-complete
codes (i.e. codes that contain all intersections of maximal codewords) are both
open convex and closed convex, and provide an upper bound for their minimal
embedding dimension. Finally, we show that the addition of non-maximal
codewords to an open convex code preserves convexity.
