Steinitz's Theorem

In polyhedral combinatorics, a branch of mathematics, Steinitz's theorem is a characterization of the undirected graphs formed by the edges and vertices of three-dimensional convex polyhedra: they are exactly the 3-vertex-connected planar graphs. That is, every convex polyhedron forms a 3-connected planar graph, and every 3-connected planar graph can be represented as the graph of a convex polyhedron. For this reason, the 3-connected planar graphs are also known as polyhedral graphs. Steinitz's theorem is named after Ernst Steinitz, who proved it in 1922. Branko Grünbaum has called this theorem “the most important and deepest known result on 3-polytopes.”

The name "Steinitz's theorem" has also been applied to other results of Steinitz:

• the Steinitz exchange lemma implying that each basis of a vector space has the same number of vectors,
• the theorem that if the convex hull of a point set contains a unit sphere, then the convex hull of a finite subset of the point contains a smaller concentric sphere, and
• Steinitz's vectorial generalization of the Riemann series theorem on the rearrangements of conditionally convergent series.