In the mathematical discipline of polyhedral combinatorics, the Gale transform turns the vertices of any convex polytope into a set of vectors or points in a space of a different dimension, the Gale diagram of the polytope. It can be used to describe high-dimensional polytopes with few vertices, by transforming them into sets with the same number of points, but in a space of a much lower dimension. The process can also be reversed, to construct polytopes with desired properties from their Gale diagrams. The Gale transform and Gale diagram are named after David Gale, who introduced these methods in a 1956 paper on neighborly polytopes.
Definitions
Transform
Given a -dimensional polytope, with vertices, adjoin 1 to the Cartesian coordinates of each vertex, to obtain a -dimensional column vector. The matrix of these column vectors has dimensions , defining a linear mapping from -space to -space, surjective with rank . The kernel of describes linear dependencies among the original vertices with coefficients summing to zero; this kernel has dimension . The Gale transform of is a matrix of dimension , whose column vectors are a chosen basis for the kernel of . Then has row vectors of dimension . These row vectors form the Gale diagram of the polytope. A different choice of basis for the kernel changes the result only by a linear transformation.
Note that the vectors in the Gale diagram are in natural bijection with the vertices of the original -dimensional polytope, but the dimension of the Gale diagram is smaller whenever .
A proper subset of the vertices of a polytope forms the vertex set of a face of the polytope, if and only if the complementary set of vectors of the Gale transform has a convex hull that contains the origin in its relative interior. Equivalently, the subset of vertices forms a face if and only if its affine span does not intersect the convex hull of the complementary vectors.
Linear diagram
Because the Gale transform is defined only up to a linear transformation, its nonzero vectors can be normalized to all be -dimensional unit vectors. The linear Gale diagram is a normalized version of the Gale transform, in which all the vectors are zero or unit vectors.
Affine diagram
Given a Gale diagram of a polytope, that is, a set of unit vectors in an -dimensional space, one can choose a -dimensional subspace through the origin that avoids all of the vectors, and a parallel subspace that does not pass through the origin. Then, a central projection from the origin to will produce a set of -dimensional points. This projection loses the information about which vectors lie above and which lie below it, but this information can be represented by assigning a sign (positive, negative, or zero) or equivalently a color (black, white, or gray) to each point. The resulting set of signed or colored points is the affine Gale diagram of the given polytope. This construction has the advantage, over the Gale transform, of using one less dimension to represent the structure of the given polytope.
Gale transforms and linear and affine Gale diagrams can also be described through the duality of oriented matroids. As with the linear diagram, a subset of vertices forms a face if and only if there is no affine function (a linear function with a possibly nonzero constant term) that assigns a non-negative value to each positive vector in the complementary set and a non-positive value to each negative vector in the complementary set.
Examples
The Gale diagram is particularly effective in describing polyhedra whose numbers of vertices are only slightly larger than their dimensions.
Simplices
A -dimensional polytope with vertices, the minimum possible, is a simplex. In this case, the linear Gale diagram is 0-dimensional, consisting only of zero vectors. The affine diagram has gray points.
One additional vertex
In a -dimensional polytope with vertices, the linear Gale diagram is one-dimensional, with the vector representing each point being one of the three numbers , , or . In the affine diagram, the points are zero-dimensional, so they can be represented only by their signs or colors without any location value. In order to represent a polytope, the diagram must have at least two points with each nonzero sign. Two diagrams represent the same combinatorial equivalence class of polytopes when they have the same numbers of points of each sign, or when they can be obtained from each other by negating all of the signs.