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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2301.12760 |
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| _version_ | 1866911169659273216 |
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| author | Maxwell, James Smith, Ben |
| author_facet | Maxwell, James Smith, Ben |
| contents | We initiate the study of convex geometry over ordered hyperfields. We define convex sets and halfspaces over ordered hyperfields, presenting structure theorems over hyperfields arising as quotients of fields. We prove hyperfield analogues of Helly, Radon and Carathéodory theorems. We also show that arbitrary convex sets can be separated via hemispaces. Comparing with classical convexity, we begin classifying hyperfields for which halfspace separation holds. In the process, we demonstrate many properties of ordered hyperfields, including a classification of stringent ordered hyperfields. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_12760 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Convex geometry over ordered hyperfields Maxwell, James Smith, Ben Metric Geometry Combinatorics Logic 52A30, 16Y20, 12J15, 52A35, 52A40 We initiate the study of convex geometry over ordered hyperfields. We define convex sets and halfspaces over ordered hyperfields, presenting structure theorems over hyperfields arising as quotients of fields. We prove hyperfield analogues of Helly, Radon and Carathéodory theorems. We also show that arbitrary convex sets can be separated via hemispaces. Comparing with classical convexity, we begin classifying hyperfields for which halfspace separation holds. In the process, we demonstrate many properties of ordered hyperfields, including a classification of stringent ordered hyperfields. |
| title | Convex geometry over ordered hyperfields |
| topic | Metric Geometry Combinatorics Logic 52A30, 16Y20, 12J15, 52A35, 52A40 |
| url | https://arxiv.org/abs/2301.12760 |