Several examples of neigbourly polyhedra in co-dimension 4
Rostislav Devyatov

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 vertices and are embedded in . Their (affine) Gale diagrams in have black points that form a convex polygon. These Gale diagams can be enumerated using 3-trees (trees with some additional structure). Given and , each of the constructed polyhedra in has a fixed number of faces of dimension that contain a vertex . (This number depends on and does not depend on the polyhedron and the vertex ).
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Taxonomy
TopicsMathematics and Applications · graph theory and CDMA systems · Advanced Combinatorial Mathematics
