  We derive an explicit formula for the evaluation of the classical closed
string action for any off-shell string field, and for the calculation of
arbitrary off-shell amplitudes. The formulae require a parametrization, in
terms of some moduli space coordinates, of the family of local coordinates
needed to insert the off-shell states on Riemann surfaces. We discuss in detail
the evaluation of the tachyon potential as a power series in the tachyon field.
The expansion coefficients in this series are shown to be geometrical
invariants of Strebel quadratic differentials whose variational properties
imply that closed string polyhedra, among all possible choices of string
vertices, yield a tachyon potential which is as small as possible order by
order in the string coupling constant. Our discussion emphasizes the
geometrical meaning of off-shell amplitudes.
