Orthant polyhedra

TL;DR

This paper classifies 2D orthant polyhedra and explores conditions under which higher-dimensional orthant polyhedra can be realized as sections of non-negative orthants, revealing that any polytope can be represented in sufficiently high dimensions.

Contribution

It provides a complete classification of 2D orthant polyhedra and advances understanding of higher-dimensional cases, showing all polytopes can be realized as orthant sections in high enough dimensions.

Findings

Complete classification of 2D orthant polyhedra

Higher-dimensional orthant polyhedra can realize any polytope

Every polytope can be represented as an orthant section in sufficiently high dimensions

Abstract

An orthant polyhedron is a polyhedron with $m$ hyperfaces, that could be realized as a section of the $m$-dimensional non-negative orthant. We classify all 2-dimensional orthant polyhedra and provide some partial results towards the classification of higher dimensional orthant polyhedra. As a consequence of our results we show that every polytope could be realized as a section of the $n$-dimensional non-negative orthant with $n$ being big enough.