  McMullen's g-vector is important for simple convex polytopes. This paper
postulates axioms for its extension to general convex polytopes. It also
conjectures that, for each dimension d, a stated finite calculation gives the
formula for the extended g-vector. This calculation is done by computer for d=5
and the results analysed. The conjectures imply new linear inequalities on
convex polytope flag vectors. Underlying the axioms is a hypothesised
higher-order homology extension to middle perversity intersection homology
(order-zero homology), which measures the failure of lower-order homology to
have a ring structure.
