Determinant of the Finite Volume Laplacian

The finite volume Laplacian can be defined in all dimensions and is a natural way to approximate the operator on simplicial mesh. In most general setting, its definition with orthogonal duals may require that not volumes are positive; an example case corresponding two-dimensional elements non-Delaunay triangulation. Nonetheless, many cases two- three-dimensional Laplacians shown negative semidefinite kernel consisting of constants. This work generalizes two gives geometric description determinant; particular, it relates determinant simplex any dimension certain quantities derived from geometry.