Intersection theorems on polytopes

Abstract

Intersection theorems on polytopesvan der Laan, G.; Talman, Dolf; Yang, Z. Publication date: 1994 Link to publication Citation for published version (APA):van der Laan, G., Talman, A. J. J., & Yang, Z. (1994). Intersection theorems on polytopes. (CentER Discussion Paper; Vol. 1994-20). Unknown Publisher. General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.-Users may download and print one copy of any publication from the public portal for the purpose of private study or research -You may not further distribute the material or use it for any profit-making activity or commercial gain -You may freely distribute the URL identifying the publication in the public portal Take down policyIf you believe that this document breaches copyright, please contact us providing details, and we will remove access to the work immediately and investigate your claim. AbstractIntersection theorcros are used to prove the existence of solutions to mathematical programming and game theoretic problems. Well-known intersection theorems are thc theorerns of Sperner, Knaster-Kuratowski-Mazurkiewicz (KKM), Scarf, Shapley, Ichiishi and Gale on the unit simplex. Recently the intersection result of KKM has been generalized by Ichiishi and Idzik to closed coverings of a compact convex polyhedron, called a pulytopc. In this paper we formulate a general intersection theorem on the polytope. To do so, we need to generalize the concept of balancedness as is used by Shapley and Ichiishi. The theorem implies most of the results stated above as special cases. First, we show that the theorems of KKM, Sperner, Scarf, Shapley and Ichiishi on the unit simplex and also some theorems of Ichiishi and Idzik on a polytope all satisfy t.he conditions of our theorem on the polytope. Secondly, the general thcurem allows us to formulate the analogs of these theorems on the polytope.

Combinatorial Lemmas for Polyhedrons

“…If we omit this condition we may still prove that there exists at least one b-balanced n-simplex (not necessarily an odd number of such n-simplexes). Related results were obtained by van der Laan, Talman and Yang [6,7]. Theorem 3.4.…”

A Polytopal Generalization of Sperner's Lemma

“…As an application of Theorem 1, we establish a version of a KKM-type intersection result of [14,Theorem 10] which now includes more cardinality information.…”

“…The surjectivity of f can also be proved as a consequence of the KKM-type result of [14,Theorem 10] Proof. First, note that because of the Sperner labelling of the triangulation T of P; the map f satisfies f ðFÞ F for any face F of P:…”

“…The non-constructive proof that we give in Section 2 relies on a known result about the surjectivity of the piecewise linear map induced by a labelled triangulation of P (Proposition 3, cf. [5,14]) and the notion of a pebble set that we develop. Pebble sets are also used in Section 2 to prove the following result of independent interest in discrete geometry.…”

Socially Structured Games

“…It will be shown that a socially stable game has a non-empty socially stable core. To do so, we formulate an intersection theorem on the unit simplex that generalizes the well-known intersection theorem used by Shapley (1973) (see also Herings, 1997;Ichiishi, 1988;van der Laan et al, 1999). Since socially stable games have a non-empty socially stable core, they also have a non-empty core.…”