Gale diagram

Summary

The Gale diagram, or Gale transform, turns $n$ points spanning $\R^d$ into $n$ vectors in $\R^{n-d-1}$. The new vectors encode the affine dependences of the old points. The faces of the convex hull of the points can be read off from them. Suppose a convex polytope has only a few more vertices than its dimension. Then its Gale diagram lives in a very low dimension and can be drawn. This makes Gale diagrams a main tool for classifying such polytopes and for building polytopes with unexpected properties.