The Tilt Formula for Generalized Simplices in Hyperbolic Space

Abstract. For a simplex in Lorentzian space whose vertices are in the positive light cone, Weeks defined the ``tilt\u27\u27 relative to each face. It gave us an efficient tool for deciding whether or not the dihedral angle between two simplices holding a face in common is convex. He also provided an efficient formula, called the ``tilt formula,\u27\u27 to obtain tilts from the intrinsic hyperbolic structure of the simplex when its dimension is two or three. Sakuma and Weeks generalized it to general dimensions. In this paper we generalize the concept of the tilt and the tilt formula to the case where not all vertices are in the positive light cone. A key to our generalization is to give a correspondence between points and hyperplanes (or half-spaces) in Lorentzian space