Following Varchenko, we consider the "analytic continuation of a
polytope". There is an explicit wall crossing formula when the polytope
changes of shape. I will also state an analogous wall crossing formula for
Duistermaat-Heckman in the context of Hamiltonian geometry.
Links:
[1] http://www.mpim-bonn.mpg.de/de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/de/node/3444
[3] http://www.mpim-bonn.mpg.de/de/node/5312