  We investigate the functional determinant of the laplacian on piece-wise flat
two-dimensional surfaces, with conical singularities in the interior and/or
corners on the boundary. Our results extend earlier investigations of the
determinants on smooth surfaces with smooth boundaries. The differences to the
smooth case are: a) different ``interaction energies'' between pairs of conical
singularities than one would expect from a naive extrapolation of the results
for a smooth surface; and b) ``self-energies'' of the singularities.
