  Much work has been done to identify which binary codes can be represented by
collections of open convex or closed convex sets. While not all binary codes
can be realized by such sets, here we prove that every binary code can be
realized by convex sets when there is no restriction on whether the sets are
all open or closed. We achieve this by constructing a convex realization for an
arbitrary code with $k$ nonempty codewords in $\mathbb{R}^{k-1}$. This result
justifies the usual restriction of the definition of convex neural codes to
include only those that can be realized by receptive fields that are all either
open convex or closed convex. We also show that the dimension of our
construction cannot in general be lowered.
