Convex geometry over ordered hyperfields
Metric Geometry
2025-09-24 v2 Combinatorics
Logic
Abstract
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 Carath\'eodory 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.
Cite
@article{arxiv.2301.12760,
title = {Convex geometry over ordered hyperfields},
author = {James Maxwell and Ben Smith},
journal= {arXiv preprint arXiv:2301.12760},
year = {2025}
}
Comments
Final version, to appear in Innovations in Incidence Geometry. 44 pages, 18 figures