The Delaunay criterion of the empty sphere is widely used for 3 dimensional tetrahedron tessellation. But original Delaunay tetrahedralization can notbe used for the points set with constrained boundary and the degenerate points set in which four ormore points are coplanar or in which five ormore points are cospherical. The concept of Delaunay tetrahedralization in an arbitrary domain (DTETAD) is presented based on the definition of local optimized triangulation which is brought out to substitute the strict empty sphere criterion of Delaunay. The sufficient and necessary condition for a tetrahedralization to be a DTETAD are proved

and the conditional empty sphere criterion ofDTETAD is presented. The research establishes the theoretic foundation for the application of Delaunay in an arbitrary domain.