The poset of rational cones

2016/06/30 by Joseph Gubeladze, Mateusz Michałek · 1 citation
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Algorithm #Combinatorics #Commutative Algebra and Its Applications #Cone (formal languages) #Conjecture #Homogenization (climate) #Mathematics #Order (exchange) #Partially ordered set #Polytope #Pure mathematics #Topological and Geometric Data Analysis #math.CO #math.NT #msc:20M13 #msc:52B20

paper · pdf · doi:10.2140/pjm.2018.292.103

published as Pacific J. Math. 292 (2018) 103-115 · Final version, to appear in Pacific Journal of Mathematics

arxiv created 2017/06/13 · openalex publication_date 2017/10/04 · arxiv updated 2018/03/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We introduce a natural partial order on the set Cones(d) of rational cones in R^d. The poset NPol(d-1) of normal polytopes in R^d-1 embeds into Cones(d) via the homogenization map. The order in Cones(d) is conjecturally the inclusion order. We prove this for d=3 and show a stronger version of the connectivity of Cones(d) for all d. Topological aspects of the conjecture are also discussed.

Citations

• Higher algebraic K-theory: I
• Polytopes, Rings, and K-Theory
• Computing the Continuous Discretely
• Graduate Texts in Mathematics

Cited by

• Convex cone [wikipedia]

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