 The "f"-vector and the "h"-vector uniquely determine each other through the relation

Then the Dehn–Sommerville equations can be restated simply as

The equations with 0 ≤ k ≤ (d−1) are independent, and the others are manifestly equivalent to them.Richard Stanley gave an interpretation of the components of the "h"-vector of a simplicial convex polytope "P" in terms of the projective toric variety "X" associated with (the dual of) "P".Namely, they are the dimensions of the even intersection cohomology groups of "X"