Combinatorial Lemmas for Polyhedrons

Abstract

We formulate general boundary conditions for a labelling to assure the existence of a balanced n-simplex in a triangulated polyhedron. Furthermore we prove a Knaster-Kuratowski-Mazurkiewicz type theorem for polyhedrons and generalize some theorems of Ichiishi and Idzik. We also formulate a necessary condition for a continuous function defined on a polyhedron to be an onto function.

Generalizing the Poincaré–Miranda theorem: the avoiding cones condition

“…For the same point x, our avoiding cones condition requires much less: only if all the other N − M components of f (x) are null, then at least one of those M inequalities must be satisfied. This shows that our Corollary 5 also generalizes [16,Theorem 3.4]. Similar considerations also apply to other variants of the Poincaré-Miranda theorem for sets D which are product of balls instead of intervals, as for instance in [21,Corollary 2].…”

“…After his pioneering work, many other researchers found applications of Theorem 1, and different generalizations have been proposed, see e.g. [5,16,21,23,24,28,29,[31][32][33][34].…”