  A convex polytope $P$ in the real projective space with reflections in the
facets of $P$ is a Coxeter polytope if the reflections generate a subgroup
$\Gamma$ of the group of projective transformations so that the
$\Gamma$-translates of the interior of $P$ are mutually disjoint. It follows
from work of Vinberg that if $P$ is a Coxeter polytope, then the interior
$\Omega$ of the $\Gamma$-orbit of $P$ is convex and $\Gamma$ acts properly
discontinuously on $\Omega$. A Coxeter polytope $P$ is $2$-perfect if $P
\smallsetminus \Omega$ consists of only some vertices of $P$.
  In this paper, we describe the deformation spaces of $2$-perfect Coxeter
polytopes $P$ of dimension $d \geqslant 4$ with the same dihedral angles when
the underlying polytope of $P$ is a truncation polytope, i.e. a polytope
obtained from a simplex by successively truncating vertices. The deformation
spaces of Coxeter truncation polytopes of dimension $d = 2$ and $d = 3$ were
studied respectively by Goldman and the third author.

Geometric Topology;Group Theory