GaleDiagramme

GaleDiagramme, often referred to simply as a Gale diagram, is a combinatorial representation used in convex geometry to study finite sets of points and the polytopes they determine. Named after the mathematician David Gale, these diagrams encode the arrangement and relationships of a polytope’s vertices in a lower-dimensional form, facilitating analysis of its face structure and realizability.

Construction and basic idea: Suppose a d-dimensional convex polytope P has n vertices v1, v2, ..., vn in general position. Form the d by n matrix V with these vertices as columns. The Gale diagram is derived from the kernel of V^T, which has dimension n − d. By selecting a basis for this kernel, one assigns to each vertex vi a corresponding vector gi in R^(n−d). The collection {gi} (often scaled to lie on a unit sphere, i.e., on S^(n−d−1)) constitutes the Gale diagram of P. In certain common cases, particularly when n = d + 2, the diagram can be drawn on a circle and analyzed with circular-arc or semicircle criteria.

Interpretation and use: Gale diagrams provide a dual viewpoint on polytopes. They encode the face lattice and help determine which subsets of vertices form faces without needing to manipulate the high-dimensional polytope directly. The diagram is also a useful tool in constructing polytopes with prescribed combinatorial types and in studying realizability problems, where one seeks polytope realizations that meet given incidences. Gale diagrams play a role in the theory of oriented matroids and in the study of neighborly and cyclic polytopes.

History and scope: The concept traces back to David Gale’s work on polyhedral combinatorics in the 1960s. Today Gale diagrams are a standard technique in discrete and computational geometry for translating high-dimensional convexity questions into lower-dimensional diagrammatic questions. See also Gale transform, convex polytope, and oriented matroids.