Let f: S → S be an operator on an n simplex with vertices {x0, x1,..., xn}. Suppose that the elements of S are labeled using the rule
x → min{i: βi ≤ αi, ≠ 0}
where αi, and βi are the barycentric coordinates of x and f (x) respectively. Show that
1. The rule assigns a label in {0, 1,..., n} to every x e S.
2. Each vertex of S retains its own label.
3. Each vertex on a face of S receives a label corresponding to one of the vertices of that face.
Hence the rule generates an admissible labeling of the simplex.