View source for Steinitz theorem

Every abstract polyhedron (cf. [[Polyhedron, abstract|Polyhedron, abstract]]) with [[Euler characteristic|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087680/s0876801.png" /> is incident with a face <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087680/s0876802.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087680/s0876803.png" /> is a part of the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087680/s0876804.png" />; a vertex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087680/s0876805.png" /> is incident with an edge <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087680/s0876806.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087680/s0876807.png" /> is an end-point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087680/s0876808.png" />; a vertex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087680/s0876809.png" /> is incident with a face <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087680/s08768010.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087680/s08768011.png" /> is one of the vertices of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087680/s08768012.png" />. 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. [[#References|[a1]]] and [[#References|[a2]]]).

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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Bokowski,   B. Sturmfels,   "Computational synthetic geometry" , ''Lect. notes in math.'' , '''1355''' , Springer  (1989)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B. Grünbaum,   "Convex polytopes" , Wiley  (1967)</TD></TR></table>