  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.
