A Polytopal Generalization of Sperner's Lemma

Abstract

We prove the following conjecture of Atanassov (Studia Sci. Math. Hungar. 32 (1996), 71-74). Let T be a triangulation of a d-dimensional polytope P with n vertices v 1 ; v 2 ; . . . ; v n : Label the vertices of T by 1; 2; . . . ; n in such a way that a vertex of T belonging to the interior of a face F of P can only be labelled by j if v j is on F: Then there are at least n À d full dimensional simplices of T; each labelled with d þ 1 different labels. We provide two proofs of this result: a non-constructive proof introducing the notion of a pebble set of a polytope, and a constructive proof using a path-following argument. Our non-constructive proof has interesting relations to minimal simplicial covers of convex polyhedra and their chamber complexes, as in

Alternate Heegaard genus bounds distance

“…The contradiction completes the proof. Elementary Proof This proof is modelled on elementary proofs of Sperner's Lemma (following very helpful comments of Francis Su, cf also [3]). As above, assume there is no such quadrilateral face, and again extend ƒ to the triangulation ƒ C as described in the first proof.…”

Lower Bounds for Simplicial Covers and Triangulations of Cubes

“…Thus f K is a PL map from P to P that takes simplices of K to simplices formed by vertices of P, and these images must be a cover of P because this map is a Brouwer map of degree 1 [3,Proposition 3]. Thus there are at least C(P) such simplices in the triangulation K .…”

“…Define the covering number C(P) to be the minimal number of simplices needed for a cover of a polytope P. Although the covering number is of interest in its own right (see [3]), we prove in Theorem 1 that the covering number of P also gives a lower bound for the size of a minimal triangulation of P, including triangulations with extra vertices.…”

Pebble Sets in Convex Polygons

“…Another obvious generalization would be to consider polytopes in higher dimensions, as in [3], replacing the notion of triangle with that of simplices. But there are many cases where the problem analogous to the polygon question has no solution.…”

“…In [3], De Loera, Peterson, and Su employ analogous sets in d-dimensional polytopes to prove a generalization of Sperner's Lemma. Following the terminology in [3], we will call a solution to the question posed in [1] a pebble set.…”