Polyhedra in spacetime from null vectors

We consider convex spacelike polyhedra oriented in the Minkowski space. These are the classical analogues of spinfoam intertwiners. We point out a parametrization of these shapes using null face normals, with no constraints or redundancies. Our construction is dimension-independent. In 3+1d, it provides the spacetime picture behind a well-known property of the loop quantum gravity intertwiner space in spinor form, namely that the closure constraint is always satisfied after some SL(2, C) rotation. As a simple application of our variables, we incorporate them in a 4-simplex action that reproduces the large-spin behavior of the Barrett-Crane vertex amplitude.