LR Characterization of Chirotopes of Finite Planar Families of Pairwise Disjoint Convex bodies
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 on the set of 3-subsets of a finite set is a chirotope of finite planar families of pairwise disjoint convex bodies if and only if for every 3-, 4-, and 5-subset of the restriction of to the set of 3-subsets of 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.
chirotopes convexity discrete geometry projective planes pseudoline arrangements