  Convex neural codes are combinatorial structures describing the intersection
pattern of a collection of convex sets. Inductively pierced codes are a
particularly nice subclass of neural codes introduced in the information
visualization literature by Stapleton et al. in 2011 and to the convex codes
literature by Gross et al. in 2016. Here, we show that all inductively pierced
codes are nondegenerate convex codes and nondegenerate hyperplane codes. In
particular, we prove that a $k$-inductively pierced code on $n$ neurons has a
convex realization with balls in $\mathbb R^{k+1}$ and with half spaces in
$\mathbb R^{n}$. We characterize the simplicial and polar complexes of
inductively pierced codes, showing the simplicial complexes are disjoint unions
of vertex decomposable clique complexes and that the polar complexes are
shellable. In an earlier version of this preprint, we gave a flawed proof that
toric ideals of $k$-inductively pierced codes have quadratic Gr\"obner bases
under the term order induced by a shelling order of the polar complex of
$\mathcal C$. We now state this as a conjecture, inspired by computational
evidence.
