Einfache, konvexe Polytope und ihre Graphen
A d-polytope is the convex hull of a finite number of affinely independent points in d-dimensional space. The faces of a d-polytope are the intersections of the d-polytope with a supporting hyperplane. A k- face is a k-dimensional face. The set of k-faces \((k=-1,...,d)\) of each d-polytope form a combinatorial complex. Two d-polytopes are combinatorially equivalent if their associated combinatorial face- complexes are isomorphic. The graph of a d-polytope is the graph formed by its 0-faces (vertices) and 1-faces (edges). It is well known that two 3-polytopes with isomorphic graphs are combinatorially equivalent. This property does not, however, generalize to higher dimensions. A d-polytope is simple if each of its vertices is contained in exactly d different d-1 faces. Its graph is then always regular of degree d. In this paper it is shown that two simple 4-polytopes with isomorphic graphs are always combinatorially equivalent.
scientific article; zbMATH DE number 3901547
scientific article