  We show that off-shell perturbative amplitudes with arbitrary number of
external lines and complex masses can be reduced to $I$-fold integrals of the
generalized Schl\"{a}fli functions, where $I$ is the number of lines in the
corresponding vacuum diagram which is independent of the number of external
lines. The Schl\"{a}fli functions are obtained as analytic continuation of the
volume of the spherical simplex as a function of the hyperplane parameters that
define the simplex. These functions have nice and thoroughly studied analytic,
geometric and number theoretic properties. They possess Gauss-Manin connection
and in conformal case are expressed by iterated integrals. Our representation
sheds new light on geometry of particle configuration spaces in perturbation
theory.
