Dehn-Sommerville equations

In geometry, Dehn-Sommerville equations are linear equations that apply to the numbers of faces of simple polytopes.

Their existence was conjectured by Max Dehn in 1905 who found them in dimension at most 5. They were discovered and proved by Duncan Sommerville in 1927.

[edit] Formulation

Let P be a d-dimensional simple polytope. Denote by f_i the number of i-dimensional faces of P, i = 0,1,..,d. Then:

$\sum_{i=k}^d (-1)^{i} \binom{i}{k} f_i \, = \, \sum_{i=d-k}^d (-1)^{d-i} \binom{i}{d-k} f_i$ for all $0\le k < \frac{d}2.$

When k=0, this equation is the Euler characteristic of a (d-1)-sphere.

There is an easier way to write these equations. Let F(t) be a generating polynomial for f_i:

$F(t) \, = \, \sum_{i=0}^d \, f_i \, t^i$

Define h-vector h_i as follows:

$F(t-1) \, = \, \sum_{i=0}^d \, h_i \, t^i$

Now the Dehn-Sommerville equations can be written as

$h_k \, = \, h_{d-k}$ for all $0\le k < d/2.$

A. Barvinok, A Course in Convexity, American Mathematical Society, Providence, 2002. ISBN 0-8218-2968-8
M. Bayer, A review of "A Course in Convexity", The American Mathematical Monthly, February 2004.
G. Ziegler, Lectures on Polytopes, Springer, 1998. ISBN 0-387-94365-X.