  The Epstein-Penner convex hull construction associates to every decorated
punctured hyperbolic surface a polyhedral convex body in the Minkowski space.
It works in the de Sitter and anti-de Sitter spaces as well. In these three
spaces, the quotient of the spacelike boundary part of the convex body has an
induced Euclidean, spherical and hyperbolic metric, respectively, with conical
singularities. We show that this gives a bijection from the decorated
Teichm\"uller space to a moduli space of such metrics in the Euclidean and
hyperbolic cases, as well as a bijection between specific subspaces of them in
the spherical case. Moreover, varying the decoration of a fixed hyperbolic
surface corresponds to a discrete conformal change of the metric. This gives a
new $3$-dimensional interpretation of discrete conformality which is in a sense
inverse to the Bobenko-Pinkall-Springborn interpretation.
