  Suppose $G$ is finitely generated group and $\mathcal{C}(G)$ consists of all
$\rho:G\to\operatorname{PGL}(n+1,\mathbb{R})$ for which there exists a properly
convex set in $\mathbb{R}\mathbb{P}^n$ that is preserved by $\rho(G)$. Then the
image of $\mathcal{C}(G)$ is closed in the character variety.
  Suppose $G$ does not contain an infinite, normal, abelian subgroup and
$\mathcal{D}(G)\subset\mathcal{C}(G)$ is the subset of holonomies of
properly-convex $n$-manifolds with fundamental group $G$. Then the image
$\mathcal{D}(G)$ is closed in the character variety.
  If $M$ is the interior of a compact $n$-manifold and $G=\pi_1M$ is as above,
and either $M$ is closed, or $\pi_1M$ contains a subgroup of infinite index
isomorphic to $\mathbb{Z}^{n-1}$, then $\mathcal{D}(G)$ is closed. If, in
addition, $M$ is the interior of a compact manifold $N$ such that every
component of $\partial N$ is $\pi_1$-injective, and finitely covered by a
torus, then every element of $\mathcal{D}(G)$ is the holonomy of a
properly-convex structure on $M$, and $\mathcal{D}(G)$ is a union of connected
components of a semi-algebraic set.
