2008/05/06 by Sven Herrmann, Herrmann, Sven, Michael Joswig +1 · 3 citations
Chemistry · Mathematics · #14M15 #52B11 #52B20 #52B30 #52B40 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG) #Synthetic Organic Chemistry Methods #math.CO #math.MG #msc:14M15 #msc:52B11 #msc:52B20 #msc:52B30 #msc:52B40
paper · pdf · doi:10.48550/arxiv.0805.0774
25 pages, 7 figures; minor corrections and changes
openalex publication_date 2008/05/06 · arxiv created 2008/07/02 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04
A split of a polytope P is a (regular) subdivision with exactly two maximal cells. It turns out that each weight function on the vertices of P admits a unique decomposition as a linear combination of weight functions corresponding to the splits of P (with a split prime remainder). This generalizes a result of Bandelt and Dress [Adv. Math. 92 (1992)] on the decomposition of finite metric spaces. Introducing the concept of compatibility of splits gives rise to a finite simplicial complex associated with any polytope P, the split complex of P. Complete descriptions of the split complexes of all hypersimplices are obtained. Moreover, it is shown that these complexes arise as subcomplexes of the tropical (pre-)Grassmannians of Speyer and Sturmfels [Adv. Geom. 4 (2004)].