On infinity type hyperplane arrangements and convex positive bijections

In this article we prove in main Theorem A that any infinity type real hyperplane arrangement \(\mathcal {H}_n^m\) with the associated normal system \(\mathcal {N}\) can be represented isomorphically by another infinity type hyperplane arrangement \(\widetilde{\mathcal {H}}_n^m\) with a given associated normal system \(\widetilde{\mathcal {N}}\) if and only if the normal systems \(\mathcal {N}\) and \(\widetilde{\mathcal {N}}\) are isomorphic, that is, there is a convex positive bijection between a pair of associated sets of normal antipodal pairs of vectors of \(\mathcal {N}\) and \(\widetilde{\mathcal {N}}\). We show in Theorem 7.1 that, if two generic hyperplane arrangements \(\mathcal {H}_n^m\) and \(\widetilde{\mathcal {H}}_n^m\) are isomorphic then their associated normal systems \(\mathcal {N}\) and \(\widetilde{\mathcal {N}}\) are isomorphic. The converse need not hold, that is, if we have two generic hyperplane arrangements \((\mathcal {H}_n^m)_1\), \((\mathcal {H}_n^m)_2\) in \(\mathbb {R}^m\), whose associated normal systems \(\mathcal {N}_1\) and \(\mathcal {N}_2\) are isomorphic, then there need not exist translates of each of the hyperplanes in the hyperplane arrangement \((\mathcal {H}_n^m)_2\), giving rise to a translated generic hyperplane arrangement \(\widetilde{\mathcal {H}}_n^m\), such that, \(\widetilde{\mathcal {H}}_n^m\) and \((\mathcal {H}_n^m)_1\) are isomorphic.