### A New Index for Polytopes

Discrete and Computational Geometry ** 6 ** (1991) 33-47.
Authors:

Andrew Klapper, 779A Anderson Hall, Dept. of Computer Science,
University of Kentucky, Lexington, KY, 40506-0046, klapper at cs.uky.edu.
www.cs.uky.edu/~klapper/andy.html

Marge Bayer, University of Kansas

**Abstract**
A new index for convex polytopes is introduced. It is a vector whose
length is the dimension of the linear span of the flag vectors of polytopes.
The existence of this index is equivalent to the generalized Dehn-Sommerville
equations. It can be computed via a shelling of the polytope.
The ranks of the middle perversity intersection homology of the
associated toric variety are computed from the index.
This gives a proof of a result of Kalai on the relationship between
the Betti numbers of a polytope and those of its dual.