  We show that the traditional criterion for a simplex to belong to the
Delaunay triangulation of a point set is equivalent to a criterion which is a
priori weaker. The argument is quite general; as well as the classical
Euclidean case, it applies to hyperbolic and hemispherical geometries and to
Edelsbrunner's weighted Delaunay triangulation. In spherical geometry, we
establish a similar theorem under a genericity condition. The weak definition
finds natural application in the problem of approximating a point-cloud data
set with a simplical complex.
