Deformation spaces of Coxeter truncation polytopes

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 $\mathrm{\Gamma }$ of the group of projective transformations so that the $\mathrm{\Gamma }$-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 $\mathrm{\Omega }$ of the $\mathrm{\Gamma }$-orbit of $P$ is convex and $\mathrm{\Gamma }$ acts properly discontinuously on $\mathrm{\Omega }$. A Coxeter polytope $P$ is $2$-perfect if $P \smallsetminus \Omega$consistsofonlysomeverticesof$P$. In this paper, we describe the deformation spaces of $2$-perfect Coxeter polytopes $P$ of dimension $d⩾4$ 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.