Geometry of ample/lopsided sets

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.

• The paper establishes a comprehensive geometric framework for ample sets, linking their combinatorial structure with intersection patterns of convex and weakly convex sets.
• It demonstrates that the cubihedra of ample sets, endowed with the intrinsic ℓ1 metric, are exactly the path-ℓ1-isometric subspaces, characterized by weak and sign-convexity.
• The study connects ample sets to oriented matroids and computational learning theory, refining combinatorial conditions like the Dress-Pajor inequality.

Geometry and Characterization of Ample/Lopsided Sets

Background and Motivation

The paper "Geometry of ample/lopsided sets" (2603.27835) establishes a comprehensive geometric framework for ample (lopsided) sets, originally introduced by Lawrence to encode intersection patterns between convex subsets of $\mathbb{R}^E$ and closed orthants. These sets have been independently rediscovered under various names—ample (Dress), extremal (Bollobás and Radcliffe), and simple (Wiedemann)—each emphasizing their deep combinatorial structure. The authors treat ample sets as isometric subgraphs of hypercubes and connect them to fundamental concepts in discrete geometry, combinatorics, and computational learning theory.

A key property is the Dress-Pajor inequality $\#\underline{X}(L) \le \#L \le \#\overline{X}(L)$, controlling the relationship between the size of the set family and its shattered/strongly shattered subsets. The paper goes beyond previous combinatorial characterizations and provides geometric and metric conditions that precisely delineate ampleness.

Geometric Realizations and Metric Structure

Ample sets $L \subseteq \{ \pm 1 \}^E$ are realized as cubihedra—cube complexes inside the hypercube $H(E) = [-1, +1]^E$. The geometric realization $|L|$ consists of all faces whose vertices belong to $L$. The connection graph $G(L)$ (the induced subgraph of the 1-skeleton $H^{(1)}(E)$) reflects the structure: $L$ is called isometric if every pair in $L$ is connected via a shortest path, and ample sets are characterized by recursive isometry and superconnectivity (isometry of all $\#\underline{X}(L) \le \#L \le \#\overline{X}(L)$0 for $\#\underline{X}(L) \le \#L \le \#\overline{X}(L)$1).

One central result is that the cubihedra of ample sets, endowed with the intrinsic $\#\underline{X}(L) \le \#L \le \#\overline{X}(L)$2 metric, are exactly the path-$\#\underline{X}(L) \le \#L \le \#\overline{X}(L)$3-isometric subspaces of $\#\underline{X}(L) \le \#L \le \#\overline{X}(L)$4. The paper introduces the notion of weak convexity (Menger-convex completeness with respect to $\#\underline{X}(L) \le \#L \le \#\overline{X}(L)$5), generalizing convexity and aligning with the metric geometry of cube complexes. Furthermore, the paper proves that weak convexity, and its stronger variant sign-convexity, provides necessary and sufficient metric and combinatorial conditions for ampleness.

Combinatorial Characterizations and Convexity Axioms

The authors enumerate a suite of equivalent conditions for ampleness, including:

• Superisometry: $\#\underline{X}(L) \le \#L \le \#\overline{X}(L)$6 is isometric for all $\#\underline{X}(L) \le \#L \le \#\overline{X}(L)$7.
• Ampleness: $\#\underline{X}(L) \le \#L \le \#\overline{X}(L)$8 (the set shattering equivalence).
• Lopsidedness: For all partitions $\#\underline{X}(L) \le \#L \le \#\overline{X}(L)$9, either $L \subseteq \{ \pm 1 \}^E$0 is strongly shattered in $L \subseteq \{ \pm 1 \}^E$1 or $L \subseteq \{ \pm 1 \}^E$2 is strongly shattered in its complement.
• Total asymmetry: Closure under antipodes of $L \subseteq \{ \pm 1 \}^E$3 in any face is trivial.

Strongly, ample sets satisfy both sparseness and maximality with respect to the Dress-Pajor inequality. The conditions also relate to recursive structures and dimension—the VC-dimension, shattered set families, and hereditary Euler characteristic.

Key metric characterizations include:

• Path-$L \subseteq \{ \pm 1 \}^E$4-isometricity: The intrinsic metric matches the ambient $L \subseteq \{ \pm 1 \}^E$5 metric.
• Sign-convexity: For any $L \subseteq \{ \pm 1 \}^E$6 differing in coordinate $L \subseteq \{ \pm 1 \}^E$7, there exists $L \subseteq \{ \pm 1 \}^E$8 satisfying $L \subseteq \{ \pm 1 \}^E$9 and matching signs elsewhere.
• 0-convexity (Signed-Circuit Axiom, SCA): For $H(E) = [-1, +1]^E$0, existence of $H(E) = [-1, +1]^E$1 with $H(E) = [-1, +1]^E$2 and $H(E) = [-1, +1]^E$3.

For subsets $H(E) = [-1, +1]^E$4, the paper proves that ampleness of $H(E) = [-1, +1]^E$5 is equivalent to 0-convexity, drawing a parallel to circuit elimination in oriented matroids but with distinct asymmetry.

Orthant Intersection Patterns and Realizability

Lawrence established that ample sets encode intersection patterns between convex sets and orthants. This work extends his result, showing that ample sets can be precisely realized as intersection patterns for weakly convex sets (path-$H(E) = [-1, +1]^E$6-isometric subsets) of $H(E) = [-1, +1]^E$7, not just convex sets. The paper further demonstrates that for every ample set there exists such a geometric realization, and conversely, every intersection pattern arising from a weakly convex subset is ample.

Additionally, commutativity between geometric realization and orthogonal projection is characterized: $H(E) = [-1, +1]^E$8 holds if and only if $H(E) = [-1, +1]^E$9 is ample, with dimension preservation under projections.

Circuits, Cocircuits, and Connections to Oriented Matroids

The paper introduces cocircuits as barycentric maps of maximal faces (facets) of cubihedra, giving a combinatorial "covector" structure analogous to oriented matroids but lacking global symmetry. The cocircuit set is precisely characterized by minimal elements under product order and adherence to the signed-circuit axiom.

Contrastingly, oriented matroids require symmetry and the existence of a zero vector. This distinction positions ample sets as a rich, asymmetric generalization within the broader framework of combinatorial convexity and cube complexes.

Implications and Developments

The geometry of ample/lopsided sets has significant ramifications in combinatorics, metric geometry, and computational learning theory (via VC-dimension and sample compression). The Dress-Pajor and Sauer-Shelah-Perles inequalities anchor the connection between learning-theoretic concepts and the geometry of cubical complexes.

Ample sets are the cubical cells in complexes of oriented matroids (COMs), which generalize both OMs and ample sets, and naturally encompass rankings and CAT(0) Coxeter zonotopal complexes. The geometric and combinatorial framework provided here deepens the understanding of intersection patterns, convexity relaxations, and their operational metrics.

Future directions may include enhanced algorithmic exploitation of weak convexity for learning and optimization, further topological analysis of cubical complexes, and exploration of ample sets within high-dimensional metric and combinatorial structures.

Conclusion

This paper delivers a rigorous geometric and metric theory for ample/lopsided sets, providing new characterizations in terms of cube complexes, weak convexity, and sign-convexity. The authors unify diverse combinatorial and geometric perspectives, link ample sets to intersection patterns of weakly convex sets, and clarify their connections and distinctions relative to oriented matroids. These results advance the structural understanding of combinatorial geometries and highlight substantial implications for both metric geometry and learning theory.