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

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The paper investigates a special class of neighbourly polytopes that live in even‑dimensional Euclidean spaces of the form (\mathbb R^{2d}) and have exactly (N = 2d+4) vertices. This situation corresponds to “codimension 4” because the number of vertices exceeds twice the ambient dimension by four. By applying Gale duality, each such polytope is represented by a Gale diagram in (\mathbb R^{2}) consisting of (d+3) black points (the “positive” points) that must form a convex polygon, together with (d+1) white points (the “negative” points). The convexity of the black points guarantees the neighbourly property: every set of at most (d) vertices spans a face.

The central contribution is a combinatorial description of all possible Gale diagrams via objects called 3‑trees. A 3‑tree is a tree whose vertices are the black points, equipped with a cyclic order at each vertex. The number of vertices of a 3‑tree is exactly (d+3). Each edge of the tree determines the placement of a white point between the two incident black points, while the cyclic orders prescribe the circular ordering of the black points in the Gale diagram. The authors prove that every admissible Gale diagram arises uniquely from a 3‑tree and vice‑versa; thus the set of neighbourly polytopes with the given parameters is in bijection with the set of 3‑trees.

Having a complete description of the Gale diagrams, the authors turn to the enumeration of faces containing a fixed vertex (A). For a given dimension (m) (with (0\le m\le d)), they show that the number of (m)-dimensional faces incident to any vertex is independent of the particular polytope or the choice of vertex. The count depends only on (d) and (m) and is given by the simple binomial formula
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