Convex geometry over ordered hyperfields

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 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.

A Result of Krasner in Categorial Form

“…In conclusion, let us note that the recent theory of enriched valuations mentioned in [25], Example 3.16 suggests further extensions of our approach, where the target of an (enriched) valuation is taken from a wider class of hyperfields than that of generalized tropical hyperfields or stringent hyperfields.…”