The poset of rational cones

TL;DR

This paper introduces a natural partial order on rational cones in R^d, explores its relation to normal polytopes, and proves key properties for the case d=3, contributing to the understanding of the structure and topology of these cones.

Contribution

It defines a new partial order on rational cones, proves the conjecture for d=3, and discusses topological aspects, advancing the theoretical understanding of cone structures.

Findings

The partial order on Cones(d) matches inclusion for d=3.

The poset NPol(d-1) embeds into Cones(d).

Connectivity properties of Cones(d) are established for all d.

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.