The Classsfication of Strongly Ismorphic Polytopes

• 1、School of Mathematics and Information Science, Guangzhou University, City 510006
• 2、School of Mathematics, Hunan University, City 410082

Abstract：In 2021, Chen-Fu-Wang introduced derived arrangements to characterize the combinatorial isomorphism classes of the parallel translations of a given hyperplane arrangement. In 2022, Fu-Wang further obtained the classification on all single-element extensions of a represented matroid (linear hyperplane arrangement). A polytope(bounded or unbounded) is closely related to a hyperplane arrangement since it can be regarded as the intersection of half-spaces determined by these hyperplanes. In this thesis, we will study all polytopes obtained from parallel translations of a given hyperplane arrangement and obtain the strongly isomorphic classification of polytopes. Listed below are our main results.Firstly, given a hyperplane arrangement with specified positive and negative half-spaces of each hyperplane, we apply the derived arrangement to study that for which parallel translations of the arrangement, the polytope obtained from the intersections of the half-spaces is non- empty,and obtain a geometric characterization on all those parallel translations. See Theorem 1 and Theorem 2. Further, we classify and characterize all such non-empty polytopes under combinatorial isomorphism. Specifically, we consider all parallel translations of a given hyperplane arrangement and the corresponding polytopes obtained from the intersections of half-spaces. If the vectors of the support number of the polytopes lie in the same face of the derived arrangement, there exist natural corresponding from face to face among such polytopes. Moreover, we obtain the isomorphism classification for all nonempty polytopes under the parallel translations of a hyperplane arrangement. See Theorem 4.

Keywords： hyperplane arrangement；polytope；derived arrangement；combinatorial isomorphism classification