  Neural codes are lists of subsets of neurons that fire together. Of
particular interest are neurons called place cells, which fire when an animal
is in specific, usually convex regions in space. A fundamental question,
therefore, is to determine which neural codes arise from the regions of some
collection of open convex sets or closed convex sets in Euclidean space. This
work focuses on how these two classes of codes -- open convex and closed convex
codes -- are related. As a starting point, open convex codes have a desirable
monotonicity property, namely, adding non-maximal codewords preserves open
convexity; but here we show that this property fails to hold for closed convex
codes. Additionally, while adding non-maximal codewords can only increase the
open embedding dimension by 1, here we demonstrate that adding a single such
codeword can increase the closed embedding dimension by an arbitrarily large
amount. Finally, we disprove a conjecture of Goldrup and Phillipson, and also
present an example of a code that is neither open convex nor closed convex.
