6.4 Gale diagrams and neighborly polytopes

Gale diagrams transform polytopes into a dual representation, simplifying analysis of high-dimensional structures. This powerful tool allows us to study facial structures and combinatorial properties, making complex geometric problems more manageable.

Neighborly polytopes are a fascinating class with unique properties. By leveraging Gale diagrams, we can construct and analyze these polytopes, uncovering their relationships to other polytope classes and exploring their existence in various dimensions.

Gale Diagrams

Gale diagrams for polytope analysis

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• Gale diagram dual representation of point configuration transforms $d$-dimensional space to $(n-d-1)$-dimensional space
• Construction process starts with $d$-dimensional polytope with $n$ vertices creates $n \times (d+1)$ matrix of homogeneous coordinates finds nullspace uses basis vectors as Gale diagram point coordinates
• Facial structure analysis uses hyperplane separation in Gale diagrams establishes correspondence between polytope faces and complements in Gale diagram
• Applications determine combinatorial properties of polytopes simplify high-dimensional geometric problems (sphere packing, linear programming)

Properties of neighborly polytopes

• Neighborly polytope every set of $\lfloor \frac{d}{2} \rfloor$ vertices forms a face
• Key properties include simplicial polytopes maximize number of faces in each dimension cyclic polytopes exemplify neighborly polytopes (hypercube, simplex)
• Relationship to other polytope classes compares with simplicial polytopes connects to centrally symmetric polytopes (cross-polytope, hypercube)

Neighborly Polytopes and Gale Diagrams

Existence of neighborly polytopes

• Gale transform method uses Gale diagrams to construct neighborly polytopes ensures no $\lfloor \frac{d}{2} \rfloor + 1$ points lie in a hyperplane
• Cyclic polytope construction uses moment curve parameterization proves neighborliness using Gale's evenness condition
• Probabilistic methods employ random point sets in high dimensions provide asymptotic existence proofs

Gale diagrams vs combinatorial properties

• Gale transform preserves combinatorial structure of polytopes
• Facial structure correspondence maps polytope faces to complements in Gale diagram $k$-faces correspond to $(n-d+k-2)$-faces in Gale diagram
• Determining $f$-vector from Gale diagram counts face numbers using hyperplane arrangements
• Simplicial polytopes and Gale diagrams characterize simplicial polytopes in Gale space (tetrahedron, octahedron)
• Applications to neighborly polytopes identify neighborly polytopes using Gale diagrams construct neighborly polytopes with specific properties (symmetry, number of vertices)