Combinatorial Analogs of Brouwer ' s Fixed Point

In this paper, we present a combinatorial theorem on a bounded polyhedron for an unrestricted integer labelling of a triangulation of the polyhedron, which can be interpreted as an extension of the Generalized Sperner lemma. When the labelling function is dual-proper, this theorem specializes to a second theorem on the polyhedron,-that is an extension of Scarf's dual Sperner lemma. These results are shown to be analogs of Brouwer's fixed point theorem on a polyhedron, and are shown to generalize two combinatorial theorems on the simplotope as well. The paper contains two other results of interest. We present a projective transformation lemma that shows that if X = (xERnJAx < e) is a bounded polyhedron, then X' = (xERni(A-eoy)x < e) is combinatorially equivalent to X if and only if y is an element of the interior of the polar of X. Secondly, the appendix contains a pseudomanifold construction for a polyhedron and its dual that may be of interest to researchers in triangulations based on primal and dual polyhedra.