Valuations on polyhedra and topological arrangements
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866914236670672896 |
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| author | Khovanskii, Askold Kiritchenko, Valentina Timorin, Vladlen |
| author_facet | Khovanskii, Askold Kiritchenko, Valentina Timorin, Vladlen |
| contents | We revisit a classical theme of (general or translation invariant) valuations on convex polyhedra. Our setting generalizes the classical one, in a ``dual'' direction to previously considered generalizations: while previous research was mostly concerned with variations of ground fields/rings, over which the vertices of polytopes are defined, we consider more general collections of defining hyperplanes. No algebraic structures are imposed on these collections. This setting allows us to uncover a close relationship between scissors congruence problems on the one hand and finite hyperplane arrangements on the other hand: there are many parallel results in these fields, for which the parallelism seems to have gone unnoticed. In particular, certain properties of the Varchenko--Gelfand algebras for arrangements translate to properties of polytope rings or valuations. Studying these properties is possible in a general topological setting, that is, in the context of the so-called topological arrangements, where hyperplanes do not have to be straight and may even have nontrivial topology. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_03176 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Valuations on polyhedra and topological arrangements Khovanskii, Askold Kiritchenko, Valentina Timorin, Vladlen Combinatorics Algebraic Topology Geometric Topology K-Theory and Homology 52B45 We revisit a classical theme of (general or translation invariant) valuations on convex polyhedra. Our setting generalizes the classical one, in a ``dual'' direction to previously considered generalizations: while previous research was mostly concerned with variations of ground fields/rings, over which the vertices of polytopes are defined, we consider more general collections of defining hyperplanes. No algebraic structures are imposed on these collections. This setting allows us to uncover a close relationship between scissors congruence problems on the one hand and finite hyperplane arrangements on the other hand: there are many parallel results in these fields, for which the parallelism seems to have gone unnoticed. In particular, certain properties of the Varchenko--Gelfand algebras for arrangements translate to properties of polytope rings or valuations. Studying these properties is possible in a general topological setting, that is, in the context of the so-called topological arrangements, where hyperplanes do not have to be straight and may even have nontrivial topology. |
| title | Valuations on polyhedra and topological arrangements |
| topic | Combinatorics Algebraic Topology Geometric Topology K-Theory and Homology 52B45 |
| url | https://arxiv.org/abs/2601.03176 |