  In a previous article, we proved tight lower bounds for the coefficients of
the generalized $h$-vector of a centrally symmetric rational polytope using
intersection cohomology of the associated projective toric variety. Here we
present a new proof based on the theory of combinatorial intersection
cohomology developed by Barthel, Brasselet, Fieseler and Kaup. This theory is
also valid for nonrational polytopes when there are no corresponding toric
varieties. So we can establish our bounds for centrally symmetric polytopes
even without requiring them to be rational.
