Cubihedra of Ample Sets

• Cubihedra of ample sets are cube complexes constructed from ample (lopsided) families in {±1}^E, defined via combinatorial and geometric equality conditions.
• They integrate combinatorial projections, Dress–Pajor inequalities, and sign-vector axioms to reveal inherent weak convexity and isometric properties in the ℓ1-metric.
• Their intrinsic ℓ1-geometry ensures canonical weakly convex realizations, linking traditional convex analysis with oriented matroid and partial cube theories.

Cubihedra of ample sets are the cube complexes canonically associated with ample, or lopsided, families $L\subseteq \{\pm1\}^E$. Given the ambient cube $H(E)=[-1,+1]^E$, the cubihedron $|L|$ is obtained by taking exactly those faces of $H(E)$ all of whose vertices belong to $L$. In this form, ampleness—initially a combinatorial equality condition in the Dress–Pajor inequalities—admits a geometric reformulation: the cubihedron $|L|$ is weakly convex in the intrinsic $\ell_1$-metric, its face barycenters form a sign-vector system in $\{\pm1,0\}^E$ satisfying an elimination axiom, and $L$ is precisely the orthant-intersection pattern of a weakly convex subset of $\mathbb R^E$ (Bandelt et al., 29 Mar 2026).

1. Origins and combinatorial definition

Ample sets were introduced under two names. Lawrence introduced lopsided sets in 1983 as subsets of $H(E)=[-1,+1]^E$0 encoding the intersection pattern of a convex set $H(E)=[-1,+1]^E$1 with the orthants of $H(E)=[-1,+1]^E$2, and Dress later called the same objects ample. In the modern formulation, one starts with a finite set $H(E)=[-1,+1]^E$3 and a family $H(E)=[-1,+1]^E$4.

Two associated simplicial complexes organize the combinatorics of $H(E)=[-1,+1]^E$5. For $H(E)=[-1,+1]^E$6, the projection family and strong fiber are

$H(E)=[-1,+1]^E$7

$H(E)=[-1,+1]^E$8

From these one defines the shattered and strongly shattered complexes

$H(E)=[-1,+1]^E$9

The Dress–Pajor inequalities are

$|L|$0

The family $|L|$1 is ample if

$|L|$2

The paper on the geometry of ample sets records several equivalent formulations. Besides the defining equality above, ampleness is equivalent to sparseness

$|L|$3

to shattering = strong shattering, to superisometry and superconnectivity of all fibers $|L|$4, to the lopsidedness condition

$|L|$5

to hereditary Euler characteristic $|L|$6 on every face intersecting $|L|$7, and to total asymmetry in Lawrence’s sense (Bandelt et al., 29 Mar 2026). These equivalences explain why ample sets appear simultaneously in VC-theory, partial cube theory, and sign-vector geometry.

2. Cubihedra inside the ambient cube

The ambient cube is

$|L|$8

Its vertices are $|L|$9, and its $H(E)$0-skeleton is the hypercube graph $H(E)$1. A face of $H(E)$2 is an $H(E)$3-fiber

$H(E)$4

where $H(E)$5. Its barycenter is the sign vector $H(E)$6 defined by

$H(E)$7

Thus zeros record free coordinates and nonzero signs record fixed coordinates.

The cubihedron of $H(E)$8 is the geometric realization

$H(E)$9

defined as the cube complex consisting of all faces $L$0 of $L$1 such that

$L$2

Its vertices are exactly the elements of $L$3, and its $L$4-skeleton is the induced subgraph

$L$5

Its maximal faces are its facets.

The barycenters of all faces form the barycentric completion

$L$6

This set is upward closed for the product order

$L$7

equivalently,

$L$8

The face poset of the cubihedron is therefore encoded by an upward-closed sign-vector system. One recovers the original family by

$L$9

The cubihedron has an explicit decomposition: $|L|$0 Hence

$|L|$1

Projection interacts tightly with ampleness. For every $|L|$2, one always has

$|L|$3

and the following are equivalent: $|L|$4 (Bandelt et al., 29 Mar 2026). In this sense, ampleness is exactly the condition that geometric realization commutes with coordinate projection.

3. Intrinsic $|L|$5-geometry

The metric setting is $|L|$6 with

$|L|$7

For $|L|$8, the $|L|$9-segment is

$\ell_1$0

an axis-parallel box.

A subset $\ell_1$1 is called weakly convex if $\ell_1$2 is complete and Menger-convex. For complete subsets of $\ell_1$3, the following are equivalent: $\ell_1$4 is weakly convex, $\ell_1$5 is a length space, and $\ell_1$6 is path-$\ell_1$7-isometric, meaning that the intrinsic metric equals the restricted ambient $\ell_1$8-metric (Bandelt et al., 29 Mar 2026).

The central metric theorem states that for $\ell_1$9, the following are equivalent:

• $\{\pm1,0\}^E$0 is ample.
• $\{\pm1,0\}^E$1 is weakly convex.
• $\{\pm1,0\}^E$2 is sign-convex.
• $\{\pm1,0\}^E$3 is sign-convex.
• $\{\pm1,0\}^E$4 is isometric and every face of $\{\pm1,0\}^E$5 is gated in $\{\pm1,0\}^E$6.
• $\{\pm1,0\}^E$7 and $\{\pm1,0\}^E$8 coincide on $\{\pm1,0\}^E$9.
• For barycenters $L$0 of any two parallel faces of $L$1,

$L$2

Thus the cubihedra of ample sets endowed with the intrinsic $L$3-metric are exactly the isometric subspaces of $L$4-spaces, called weakly convex sets in the paper.

A standard counterexample shows why graph isometry alone is insufficient. Let $L$5 and let $L$6 be all vertices of $L$7 except the two constant sign vectors. Then $L$8 is an isometric $L$9-cycle, but $\mathbb R^E$0 is not ample. Its cubihedron $\mathbb R^E$1 is a solid hexagon in $\mathbb R^E$2, and opposite midpoints have ambient $\mathbb R^E$3-distance $\mathbb R^E$4 but intrinsic distance $\mathbb R^E$5 (Bandelt et al., 29 Mar 2026). The failure is therefore cubical and metric, not merely graph-theoretic.

4. Barycenter maps, cocircuits, and sign-vector axioms

The barycenter description of a cubihedron leads to a sign-vector theory parallel to the covector language of oriented matroids. For a subset $\mathbb R^E$6, sign-convexity requires that whenever $\mathbb R^E$7 and $\mathbb R^E$8, there exists $\mathbb R^E$9 with $H(E)=[-1,+1]^E$00 and compatible signs on coordinates where $H(E)=[-1,+1]^E$01. A weaker condition is $H(E)=[-1,+1]^E$02-convexity, where for $H(E)=[-1,+1]^E$03,

$H(E)=[-1,+1]^E$04

For $H(E)=[-1,+1]^E$05, $H(E)=[-1,+1]^E$06-convexity is equivalent to the signed-circuit axiom (SCA): $H(E)=[-1,+1]^E$07 such that

$H(E)=[-1,+1]^E$08

For upward-closed subsets of $H(E)=[-1,+1]^E$09, sign-convexity and $H(E)=[-1,+1]^E$10-convexity are equivalent.

This yields a full characterization of barycenter systems. For $H(E)=[-1,+1]^E$11 and

$H(E)=[-1,+1]^E$12

the following are equivalent: $H(E)=[-1,+1]^E$13 is weakly convex; $H(E)=[-1,+1]^E$14 is sign-convex; $H(E)=[-1,+1]^E$15 is $H(E)=[-1,+1]^E$16-convex; $H(E)=[-1,+1]^E$17 is $H(E)=[-1,+1]^E$18-convex; $H(E)=[-1,+1]^E$19 is an isometric subset of the grid $H(E)=[-1,+1]^E$20; and $H(E)=[-1,+1]^E$21 is ample with

$H(E)=[-1,+1]^E$22

(Bandelt et al., 29 Mar 2026). Accordingly, a collection of $H(E)=[-1,+1]^E$23-sign vectors arises as the full barycenter set of an ample cubihedron exactly when it satisfies the elimination law encoded by (SCA).

The barycenters of the maximal faces are the cocircuits

$H(E)=[-1,+1]^E$24

The paper proves that the following are equivalent: $H(E)=[-1,+1]^E$25 is ample; $H(E)=[-1,+1]^E$26 satisfies (SCA); $H(E)=[-1,+1]^E$27 satisfies (SCA); $H(E)=[-1,+1]^E$28 satisfies (SCA); and $H(E)=[-1,+1]^E$29 satisfies (SCA). Moreover, if $H(E)=[-1,+1]^E$30 is pairwise minimal and satisfies (SCA), then

$H(E)=[-1,+1]^E$31

is ample and

$H(E)=[-1,+1]^E$32

This is the direct analogue of reconstructing oriented matroids from cocircuits, except that the global symmetry $H(E)=[-1,+1]^E$33 is absent.

5. Realizability by weakly convex sets

For $H(E)=[-1,+1]^E$34, define

$H(E)=[-1,+1]^E$35

equivalently, $H(E)=[-1,+1]^E$36 iff the closed orthant

$H(E)=[-1,+1]^E$37

meets $H(E)=[-1,+1]^E$38. Lawrence showed that if $H(E)=[-1,+1]^E$39 is convex, then $H(E)=[-1,+1]^E$40 is ample, but the paper emphasizes that not every ample set is realizable by an ordinary convex set (Bandelt et al., 29 Mar 2026).

The realizability theorem replaces convexity by weak convexity: $H(E)=[-1,+1]^E$41 Equivalently, one may take $H(E)=[-1,+1]^E$42 to be a compact weakly convex subset of $H(E)=[-1,+1]^E$43. The canonical realizing set is the cubihedron itself: $H(E)=[-1,+1]^E$44 Thus every ample set is an orthant-intersection pattern, but the correct ambient notion is weak convexity rather than Euclidean convexity.

This result places cubihedra at the center of the realization theory. They are not merely combinatorial shadows of ample sets; they furnish universal weakly convex realizations. A common misconception is therefore that convex realizability is the fundamental geometric model. The literature shows instead that convex realizability is stricter, while cubihedra recover the entire class once one passes to intrinsic $H(E)=[-1,+1]^E$45-geometry (Bandelt et al., 29 Mar 2026).

6. Relation to COMs, partial cubes, and later developments

The cubihedral viewpoint connects ample sets to the broader theory of sign-vector systems. The geometry paper states that later work on COMs (complexes of oriented matroids) generalizes both oriented matroids and ample sets, and that ample sets are exactly the COMs whose cells are cubes (Bandelt et al., 29 Mar 2026). In this formulation, the cubihedron $H(E)=[-1,+1]^E$46 is the all-cubes case of a more general cell-complex theory.

A complementary result concerns completions. The paper on ample completions proves that if $H(E)=[-1,+1]^E$47 is an oriented matroid of rank $H(E)=[-1,+1]^E$48, or more generally a complex of uniform oriented matroids with tope graph of VC-dimension $H(E)=[-1,+1]^E$49, then its tope graph can be completed to an ample partial cube of the same VC-dimension (Chepoi et al., 2020). The construction proceeds by completing oriented matroids to uniform oriented matroids of the same rank, then recursively completing uniform oriented matroid tope graphs to ample partial cubes; for complexes of uniform oriented matroids, one completes the facets and glues the resulting ample pieces along gated overlaps. The paper also conjectures that every COM can be completed to an ample partial cube without increasing VC-dimension (Chepoi et al., 2020).

These completion results are not statements about cubihedra of ample sets in the narrow sense, but they show that ample cube complexes function as canonical cubical envelopes for broader sign-vector geometries. A plausible implication is that the cubihedron of an ample set is both a terminal model for the geometry of that ample set and a target object for dimension-preserving completion procedures starting from oriented-matroidal data. In this sense, the theory supports the conclusion stated in the geometry paper: the concept of ample sets is quite natural in the context of cube complexes (Bandelt et al., 29 Mar 2026).