The Order Dimension of Convex Polytopes

1993/05/01 by Graham Brightwell, William T. Trotter · 38 citations
Computer Science · Engineering · Mathematics · #Combinatorics #Convex optimization #Convex polytope #Convex set #Dimension (graph theory) #Geometry #Graph Labeling and Dimension Problems #Limits and Structures in Graph Theory #Mathematics #Order (exchange) #Partially ordered set #Polytope #Regular polygon #graph theory and CDMA systems

paper · doi:10.1137/0406018

published in SIAM Journal on Discrete Mathematics 6(2), 230-245 (Society for Industrial and Applied Mathematics)

openalex publication_date 1993/05/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31

Abstract

With a convex polytope M in ℝ³, a partially ordered set P_M is associated whose elements are the vertices, edges, and faces of M ordered by inclusion. This paper shows that the order dimension of P_M is exactly 4 for every convex polytope M. In fact, the subposet of P_M determined by the vertices and faces is critical in the sense that deleting any element leaves a poset of dimension 3.

Citations

• Planar graphs and poset dimension
• Partially Ordered Sets
• Minimal scrambling sets of simple orders

Cited by

• Geometric Realization of Simplicial Complexes
• The Order Dimension of Planar Maps
• Representations of Planar Graphs
• Schnyder's theorem [wikipedia]