  We define a set theoretic "analytic continuation" of a polytope defined by
inequalities. For the regular values of the parameter, our construction
coincides with the parallel transport of polytopes in a mirage introduced by
Varchenko. We determine the set-theoretic variation when crossing a wall in the
parameter space, and we relate this variation to Paradan's wall-crossing
formulas for integrals and discrete sums. As another application, we refine the
theorem of Brion on generating functions of polytopes and their cones at
vertices. We describe the relation of this work with the equivariant index of a
line bundle over a toric variety and Morelli constructible support function.
