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.
