The poset of rational cones

Joseph Gubeladze and Mateusz Michałek

Vol. 292 (2018), No. 1, 103–115

DOI: 10.2140/pjm.2018.292.103

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

Milestones

Received: 14 June 2016

Revised: 29 May 2017

Accepted: 8 June 2017

Published: 22 September 2017