Several examples of neigbourly polyhedra in co-dimension 4

TL;DR

This paper constructs a series of neighborly polyhedra in 2d-dimensional space with specific vertex counts, analyzes their Gale diagrams, and explores their face structure, providing new combinatorial insights into high-dimensional polyhedral geometry.

Contribution

It introduces a new family of neighborly polyhedra in co-dimension 4 with explicit Gale diagram enumeration and face count properties, advancing understanding of high-dimensional polyhedral structures.

Findings

Constructed neighborly polyhedra with 2d+4 vertices in R^{2d}

Gale diagrams can be enumerated using 3-trees

Number of faces containing a vertex is fixed and depends only on d and m

Abstract

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 $A$).