Dehn–Sommerville equations

Remarks

One proof uses a shelling of the boundary complex, an order of the facets in which each new facet meets the earlier ones in a union of its own facets. For each facet, record the size of its smallest new face; then $h_i$ is the number of facets for which this size is $i$. The reverse of a line shelling is again a shelling, and reversing the order turns size $i$ into size $d - i$, which gives the symmetry.

The equation $h_0 = h_d$ is equivalent to Euler's relation $f_0 - f_1 + f_2 - \dots + (-1)^{d-1} f_{d-1} = 1 - (-1)^d$ for the boundary. For $d = 3$ the equations give $f_1 = 3 f_0 - 6$ and $f_2 = 2 f_0 - 4$, so the number of vertices determines the whole f-vector of a simplicial $3$-polytope.

Example: the octahedron has $h = (1, 3, 3, 1)$, which is symmetric.

The hypothesis that the complex is the boundary of a polytope (more generally, a sphere) is necessary. The full triangle, viewed as a simplicial complex of dimension $2$, has $f = (1, 3, 3, 1)$ and $h = (1, 0, 0, 0)$, which is not symmetric.

Going deeper

For a student

The equations hold more generally for simplicial spheres and for Eulerian complexes. They show that about half of the entries of the f-vector of a simplicial $d$-polytope determine the rest. The remaining question, which h-vectors occur at all, is answered by the g-theorem.