2013/06/25 by Richard Cowan, Cowan, Richard, V. Weiß +2
Computer Science · Mathematics · #05B45 #51M20 #52B10 #52C17 #60D05 #60G55 #Computational Geometry and Mesh Generation #Digital Image Processing Techniques #FOS: Mathematics #Point processes and geometric inequalities #Probability (math.PR) #math.PR #msc:05B45 #msc:51M20 #msc:52B10 #msc:52C17 #msc:60D05 #msc:60G55
paper · pdf · doi:10.48550/arxiv.1306.5862
29 pages, 38 figures
arxiv created 2013/06/25 · openalex publication_date 2013/06/25 · arxiv updated 2013/06/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04
Tessellations of R3 that use convex polyhedral cells to fill the space can be extremely complicated, especially if they are not facet-to-facet, that is, if the facets of a cell do not necessarily coincide with the facets of that cell's neighbours. In a recent paper (Weiss and Cowan, Adv.Appl.Prob. 2011), we have developed a theory which covers these complicated cases, at least with respect to their combinatorial topology. The theory required seven parameters, three of which suffice for facet-to-facet cases; the remaining four parameters are needed for the awkward adjacency concepts that arise in the general case. This current paper establishes constraints that apply to these seven parameters and so defines a permissible region within their seven-dimensional space, a region which we discover is not bounded. Our constraints in the relatively simple facet-to-facet case are also new.