Deformed Intersections of Half-spaces

TL;DR

This paper investigates the classification of extended deformations of convex polyhedra and hyperplane arrangements, establishing equivalences based on derived arrangements and exploring three types of deformations.

Contribution

It introduces a framework linking extended deformations of convex polyhedra and hyperplane arrangements through derived arrangements, providing new characterizations of their equivalence relations.

Findings

Extended deformations are normally equivalent if parameterized by the same face of the derived arrangement.

Characterization of three deformations: parallel translations, conings, and elementary lifts.

New descriptions of real derived arrangements related to faces and sign vectors.

Abstract

This paper is devoted to the classification problems concerning extended deformations of convex polyhedra and real hyperplane arrangements in the following senses: combinatorial equivalence of face posets, normal equivalence on normal fans of convex polyhedra, and sign equivalence on half-spaces. The extended deformations of convex polyhedra arise from parallel translations of given half-spaces and hyperplanes, whose normal vectors give rise to the so-called ``derived arrangement'' proposed by Rota as well as Crapo in different forms. We show that two extended deformations of convex polyhedra are normally (combinatorially, as a consequence) equivalent if they are parameterized by the same open face of the derived arrangement. Note that these extended deformations are based on parallel translations of the given hyperplanes. It allows us to study three deformations of real hyperplane arrangements: parallel translations, conings, and elementary lifts, whose configuration spaces are parameterized by open faces of the derived arrangement. Consequently, it gives a characterization of the normal, combinatorial, and sign equivalences of those three deformations via the derived arrangement. Additionally, the relationships among these three equivalence relations are discussed, and several new descriptions of real derived arrangements associated with faces and sign vectors of real hyperplane arrangements are provided.