One-loop integrals from volumes of orthoschemes

Recently in arXiv:2012.05599 Rudenko presented a formula for the volume of hyperbolic orthoschemes in terms of alternating polylogarithms. We use this result to provide an explicit analytic result for the one-loop scalar n-gon Feynman integral in n dimensions, for even n, with massless or massive internal and external edges. Furthermore, we evaluate the general six-dimensional hexagon integral in terms of classical polylogarithms.