LR characterization of chirotopes of finite planar families of pairwise disjoint convex bodies

TL;DR

This paper generalizes the classical LR characterization of chirotopes from point sets to disjoint convex bodies in the plane, introducing arrangements of double pseudolines via polarity maps.

Contribution

It extends the LR characterization to convex bodies and introduces arrangements of double pseudolines as a new conceptual tool.

Findings

Characterization of chirotopes for disjoint convex bodies

Introduction of arrangements of double pseudolines

Use of polarity maps as a key tool

Abstract

We extend the classical LR characterization of chirotopes of finite planar families of points to chirotopes of finite planar families of pairwise disjoint convex bodies: a map \c{hi} on the set of 3-subsets of a finite set I is a chirotope of finite planar families of pairwise disjoint convex bodies if and only if for every 3-, 4-, and 5-subset J of I the restriction of \c{hi} to the set of 3-subsets of J is a chirotope of finite planar families of pairwise disjoint convex bodies. Our main tool is the polarity map, i.e., the map that assigns to a convex body the set of lines missing its interior, from which we derive the key notion of arrangements of double pseudolines, introduced for the first time in this paper.