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.
vertices, adjoin 1 to the Cartesian coordinates of each vertex, to obtain a
original vertices with coefficients summing to zero; this kernel has dimension
, whose column vectors are a chosen basis for the kernel of
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.
[2] Note that the vectors in the Gale diagram are in natural bijection with the
-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.
through the origin that avoids all of the vectors, and a parallel subspace
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.
[6] 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.
[7] The Gale diagram is particularly effective in describing polyhedra whose numbers of vertices are only slightly larger than their dimensions.
In this case, the linear Gale diagram is 0-dimensional, consisting only of zero vectors.
vertices, the linear Gale diagram is one-dimensional, with the vector representing each point being one of the three numbers
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.
, the only possibility is two points of each nonzero sign, representing a convex quadrilateral.
[8] In general, the number of distinct Gale diagrams with
vertices, it is not completely trivial to determine when two Gale diagrams represent the same polytope.
[8] Three-dimensional polyhedra with six vertices provide natural examples where the original polyhedron is of a low enough dimension to visualize, but where the Gale diagram still provides a dimension-reducing effect.
Gale diagrams have been used to provide a complete combinatorial enumeration of the
vertices,[11] and to construct polytopes with unusual properties.