  A Delaunay decomposition is a cell decomposition in R^d for which each cell
is inscribed in a Euclidean ball which is empty of all other vertices. This
article introduces a generalization of the Delaunay decomposition in which the
Euclidean balls in the empty ball condition are replaced by other families of
regions bounded by certain quadratic hypersurfaces. This generalized notion is
adaptable to geometric contexts in which the natural space from which the point
set is sampled is not Euclidean, but rather some other flat semi-Riemannian
geometry, possibly with degenerate directions. We prove the existence and
uniqueness of the decomposition and discuss some of its basic properties. In
the case of dimension d = 2, we study the extent to which some of the
well-known optimality properties of the Euclidean Delaunay triangulation
generalize to the higher signature setting. In particular, we describe a higher
signature generalization of a well-known description of Delaunay decompositions
in terms of the intersection angles between the circumscribed circles.
