Convex geometry over ordered hyperfields

TL;DR

This paper explores convex geometry over ordered hyperfields, establishing fundamental theorems and properties that extend classical convexity concepts to this algebraic setting.

Contribution

It introduces convex sets and halfspaces over ordered hyperfields, proves hyperfield analogues of key convexity theorems, and classifies hyperfields based on separation properties.

Findings

Hyperfield versions of Helly, Radon, and Carathéodory theorems established.

Convex sets can be separated via hemispaces in hyperfield context.

Classification of stringent ordered hyperfields provided.

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.