This paper defines new intersection homology groups. The basic idea is this. Ordinary homology is locally trivial. Intersection homology is not. It may have significant local cycles. A localglobal cycle is defined to be a family of such local cycles that is, at the same time, a global cycle. The motivating problem is the numerical characterisation of the flag vectors of convex polytopes. Centra...

We characterize C embedded hypersurfaces of R as the only locally closed sets with continuously varying flat tangent cones whose measuretheoretic-multiplicity is at most m < 3/2. It follows then that any (topological) hypersurface which has flat tangent cones and is supported everywhere by balls of uniform radius is C. In the real analytic case the same conclusion holds under the weakened hypot...