Vol. 292, No. 1, 2018

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The poset of rational cones

Vol. 292 (2018), No. 1, 103–115
Abstract

We introduce a natural partial order on the set $Cones\left(d\right)$ of rational cones in ${ℝ}^{d}$. The poset of normal polytopes, studied by Bruns and the authors (Discrete Comput. Geom. 56:1 (2016), 181–215), embeds into $Cones\left(d\right)$ via the homogenization map. The order in $Cones\left(d\right)$ is conjecturally the inclusion order. We prove this for $d=3$ and show a stronger version of the connectivity of $Cones\left(d\right)$ for all $d$. Topological aspects of the conjecture are also discussed.

Keywords
rational cone, poset of cones, geometric realization of a poset
Mathematical Subject Classification 2010
Primary: 52B20
Secondary: 05E99, 20M13, 52C07