On infinity type hyperplane arrangements and convex positive bijections

C. P. Anil Kumar ()
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C. P. Anil Kumar: Technology and Policy # 18 & #19

Indian Journal of Pure and Applied Mathematics, 2025, vol. 56, issue 3, 1235-1259

Abstract: Abstract In this article we prove in main Theorem A that any infinity type real hyperplane arrangement $$\mathcal {H}_n^m$$ H n m with the associated normal system $$\mathcal {N}$$ N can be represented isomorphically by another infinity type hyperplane arrangement $$\widetilde{\mathcal {H}}_n^m$$ H ~ n m with a given associated normal system $$\widetilde{\mathcal {N}}$$ N ~ if and only if the normal systems $$\mathcal {N}$$ N and $$\widetilde{\mathcal {N}}$$ N ~ are isomorphic, that is, there is a convex positive bijection between a pair of associated sets of normal antipodal pairs of vectors of $$\mathcal {N}$$ N and $$\widetilde{\mathcal {N}}$$ N ~ . We show in Theorem 7.1 that, if two generic hyperplane arrangements $$\mathcal {H}_n^m$$ H n m and $$\widetilde{\mathcal {H}}_n^m$$ H ~ n m are isomorphic then their associated normal systems $$\mathcal {N}$$ N and $$\widetilde{\mathcal {N}}$$ N ~ are isomorphic. The converse need not hold, that is, if we have two generic hyperplane arrangements $$(\mathcal {H}_n^m)_1$$ ( H n m ) 1 , $$(\mathcal {H}_n^m)_2$$ ( H n m ) 2 in $$\mathbb {R}^m$$ R m , whose associated normal systems $$\mathcal {N}_1$$ N 1 and $$\mathcal {N}_2$$ N 2 are isomorphic, then there need not exist translates of each of the hyperplanes in the hyperplane arrangement $$(\mathcal {H}_n^m)_2$$ ( H n m ) 2 , giving rise to a translated generic hyperplane arrangement $$\widetilde{\mathcal {H}}_n^m$$ H ~ n m , such that, $$\widetilde{\mathcal {H}}_n^m$$ H ~ n m and $$(\mathcal {H}_n^m)_1$$ ( H n m ) 1 are isomorphic.

Keywords: Linear inequalities in many variables; Hyperplane arrangements; Infinity type hyperplane arrangements; Primary:; 52C35 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s13226-024-00583-7

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