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