Geometry of ample/lopsided sets

2026-03-29

AI Summary

This work investigates the geometric realization of ample (lopsided) sets and their intrinsic connection to isometric subspaces of ℓ₁ space. By introducing cubihedra, the paper establishes correspondences between ample sets and weakly convex sets, intersections of coordinate-orthant regions, and systems of {±1,0}-valued sign vectors. It provides the first equivalent characterization of when an ample set admits an isometric embedding into ℓ₁ space and, drawing an analogy with oriented matroid theory, proposes a novel combinatorial representation based on sign vectors. The results show that every ample set can be expressed as the intersection of a weakly convex set with a coordinate-orthant region, thereby yielding multiple complementary geometric and combinatorial characterizations.

Abstract

Lopsided sets were introduced by Jim Lawrence in 1983 when he studied the subsets of $\{-1,+1\}^E$ that encode the intersection pattern of a convex set $K$ with the orthants of ${\mathbb R}^E$. Lopsided sets have been independently rediscovered by several other authors, in particular by Andreas Dress in 1995, who called them \emph{ample} sets. Dress defined ample sets as the set families satisfying equality in a combinatorial inequality, which holds for all set families. In a previous article we characterized ample sets in various combinatorial and graph-theoretical ways. In this paper we study geometric realizations of ample sets as cubihedra (cube complexes), which yields several new characterizations. One such characterization establishes that the cubihedra of ample sets endowed with the intrinsic $\ell_1$-metric are exactly the isometric subspaces of $\ell_1$-spaces (which we call, weakly convex sets). We also view the barycenter maps of faces of cubihedra of ample sets as collections of $\{ \pm 1, 0\}$-sign vectors and, in analogy with the characterization of oriented matroids by the covectors and the cocircuits. Moreover, we characterize the collections of $\{ \pm 1, 0\}$-sign vectors corresponding to barycenter maps of all faces and all maximal faces of an ample set. Furthermore, we show that any ample set $\covectors\subseteq \{ -1,+1\}^E$ is realizable as the intersection pattern of a weakly convex set $K$ with the orthants of ${\mathbb R}^E$. All this testifies that the concept of ample sets is quite natural in the context of cube complexes.

Problem

• ample sets
• lopsided sets
• cubihedra
• isometric subspaces
• sign vectors

Innovation

• ample sets
• lopsided sets
• cubihedra
• ℓ₁-isometric subspaces
• sign vectors