On Infinity Type Hyperplane Arrangements and Convex Positive Bijections

Abstract

In this article we prove in main Theorem A that any infinity type real hyperplane arrangement $\mathcal{H}_n^m$ (Definition 2.11) with the associated normal system $\mathcal{N}$ (Definitions [2.2,2.4] can be represented isomorphically (Definition 2.6) by another infinity type hyperplane arrangement $\tilde{\mathcal{H}}_n^m$ with a given associated normal system $\tilde{\mathcal{N}}$ if and only if the normal systems $\mathcal{N}$ and $\tilde{\mathcal{N}}$ are isomorphic, that is, there is a convex positive bijection (Definition 2.5) between a pair of associated sets of normal antipodal pairs of vectors of $\mathcal{N}$ and $\tilde{\mathcal{N}}$.