Steinitz theorem

Every abstract polyhedron (cf. Polyhedron, abstract) with Euler characteristic equal to 2 can be realized as a convex polyhedron. Here, an abstract polyhedron means a finite set of arbitrary elements, called vertices, edges and faces, for which a symmetric and transitive incidence relation is defined: An edge $a$ is incident with a face $\alpha$ if $a$ is a part of the boundary of $\alpha$; a vertex $A$ is incident with an edge $a$ if $A$ is an end-point of $a$; a vertex $A$ is incident with a face $\alpha$ if $A$ is one of the vertices of $\alpha$. The system of vertices, edges and faces of an abstract polyhedron should satisfy the following conditions:

1) Each edge is incident with two and only two vertices. Each edge is incident with two and only two faces.

2) Two vertices can have only one edge incident with both of them. Two faces can have only one edge incident with both of them.

3) Each vertex is incident with at least three faces. Each vertex is incident with at least three edges.

The theorem was proved by E. Steinitz in 1917.

Comments

There are many partial results in higher dimensions and on related problems (cf. e.g. [a1] and [a2]).

It has been shown that for higher dimensions no direct analogue of the Steinitz theorem can be expected (cf. [a1]).

References

[a1]	J. Bokowski, B. Sturmfels, "Computational synthetic geometry" , Lect. notes in math. , 1355 , Springer (1989)
[a2]	B. Grünbaum, "Convex polytopes" , Wiley (1967)