  We summarize two geometrical approaches to analytically evaluate higher-fold
Mellin-Barnes (MB) integrals in terms of hypergeometric functions. The first
method is based on intersections of conic hulls, while the second one, which is
more recent, relies on triangulations of a set of points. We demonstrate that,
once automatized, the triangulation approach is computationally more efficient
than the conic hull approach. As an application of this triangulation approach,
we describe how one can derive simpler hypergeometric solutions of the
conformal off-shell massless two-loop double box and one-loop hexagon Feynman
integrals than those previously obtained from the conic hull approach. Lastly,
by applying the above techniques on the MB representation of multiple
polylogarithms, we show how to obtain new convergent series representations for
these functions. These new analytic expressions were numerically cross-checked
with GINAC.
