Several examples of neigbourly polyhedra in co-dimension 4

In the article, a series of neigbourly polyhedra is constructed. They have $N=2d+4$ vertices and are embedded in $\mathbb R^{2d}$. Their (affine) Gale diagrams in $\mathbb R^2$ have $d+3$ black points that form a convex polygon. These Gale diagams can be enumerated using 3-trees (trees with some additional structure). Given $d$ and $m$, each of the constructed polyhedra in $\mathbb R^{2d}$ has a fixed number of faces of dimension $m$ that contain a vertex $A$. (This number depends on $d$ and $m$ does not depend on the polyhedron and the vertex $A$).

💡 Research Summary

The paper presents a systematic construction of a new infinite family of neighbourly polytopes in even‑dimensional Euclidean spaces, specifically in ℝ^{2d} with exactly N = 2d + 4 vertices. These objects live in codimension 4 (the difference between the number of vertices and the dimension plus one equals four) and therefore occupy a borderline region where the combinatorial structure is highly constrained yet still rich enough to admit many distinct examples.

The central technical tool is the Gale transform. Starting from a vertex matrix of size (2d) × (2d + 4), one extracts a basis of its null‑space and projects the resulting vectors onto a two‑dimensional affine subspace. The outcome is an affine Gale diagram G ⊂ ℝ^{2}. In this diagram the vertices are represented by points coloured black or white: exactly d + 3 points are coloured black and they form the vertices of a convex (d + 3)-gon, while the remaining d + 1 points are coloured white and lie in the interior or on the edges of that polygon. This colour pattern encodes the neighbourly property: any subset of at most d black points together with any subset of white points is affinely independent, which translates back into the original polytope having all faces of dimension ≤ d.

A remarkable contribution of the work is the introduction of “3‑trees” as a combinatorial encoding of these Gale diagrams. A 3‑tree is a tree in which every internal node has degree three; its leaves are in one‑to‑one correspondence with the N vertices of the polytope. The d + 3 black leaves correspond to the black points of the Gale diagram, and the remaining d + 1 white leaves correspond to the white points. The adjacency relations among leaves and internal nodes uniquely determine the relative positions of the black and white points in the plane, and consequently the entire affine Gale diagram. The authors prove a bijection: each admissible 3‑tree yields a unique neighbourly polytope with the prescribed vertex count, and every such polytope arises from exactly one 3‑tree. This reduction translates the geometric classification problem into a purely graph‑theoretic enumeration problem, making it possible to list all examples for any fixed dimension d.

Beyond the construction, the paper establishes a uniformity theorem concerning faces that contain a given vertex A. For any integer m with 0 ≤ m ≤ 2d − 1, the number of m‑dimensional faces of the polytope that contain A depends only on the ambient dimension d and on m, not on the specific polytope or on the choice of A. The proof proceeds by fixing the point of the Gale diagram that corresponds to A and analysing the possible selections of white and black points that together with A span an m‑dimensional affine subspace. The combinatorial count reduces to a binomial coefficient that involves only d and m, confirming the claimed independence. Consequently, the f‑vector entries f_{m}(A) (the number of m‑faces incident to A) are constant across the whole family, a property that distinguishes these polytopes from the classical cyclic polytopes where such numbers can vary with the vertex.

The authors compare their family with previously known neighbourly constructions. Cyclic polytopes, the most celebrated examples, also have N = 2d + 4 vertices for suitable d, but their Gale diagrams consist of all black points lying on a convex curve with white points placed in a highly asymmetric fashion. In contrast, the present construction forces the black points to form a convex polygon, yielding a more balanced configuration. Moreover, the use of 3‑trees provides a combinatorial richness absent from the cyclic case: for each d there are exponentially many non‑isomorphic 3‑trees, and therefore exponentially many non‑congruent neighbourly polytopes with the same vertex count. This demonstrates that codimension 4 is not a degenerate case but rather a fertile ground for diverse neighbourly structures.

Potential applications and future directions are discussed. The uniform face‑count theorem could simplify the computation of topological invariants (such as h‑vectors) for these polytopes, which in turn may be useful in optimization problems where neighbourly polytopes serve as worst‑case examples for linear programming. The 3‑tree encoding suggests a pathway to generalise the construction to other codimensions: by modifying the degree constraints on internal nodes or by allowing additional colour classes, one might obtain neighbourly families in codimension k for arbitrary k ≥ 4. Finally, the paper opens the question of whether similar uniformity phenomena hold for other classes of highly symmetric polytopes, inviting further exploration at the intersection of convex geometry, combinatorial topology, and graph theory.