منابع مشابه
The LSB Theorem Implies the KKM Lemma
Let S be the unit d-sphere, the set of all points of unit Euclidean distance from the origin in R . Any pair of points in S of the form x,−x is a pair of antipodes in S. Let ∆ be the d-simplex formed by the convex hull of the standard unit vectors in R. Equivalently, ∆ = {(x1, ..., xd+1) : ∑ i xi = 1, xi ≥ 0}. The following are two classical results about closed covers of these topological spaces:
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Kakutani’s fixed point theorem has many applications in economics and game theory. One of its most well-known applications is in John Nash’s paper [8], where the theorem is used to prove the existence of an equilibrium strategy in n-person games. Sperner’s lemma, on the other hand, is a combinatorial result concerning the labelling of the vertices of simplices and their triangulations. It is kn...
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In this review, firstly, we recall Ky Fan’s contributions to the KKM theory based on his celebrated 1961 KKM lemma (or the Fan-KKM theorem). Secondly, we introduce relatively recent applications of the Fan lemma due to other authors in the 21st century. Finally, some historical remarks on related works are added.
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In the summer of 2004, I participated in an REU program at East Tennessee State University. Many of the investigations I worked on were successful, leading to 4-6 papers, some of which have been submitted, and others are still in the writing process. There are many unanswered questions left in the investigations that I was conducting, and some of the work already done is not complete. Given tha...
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The partial KKM principle for an abstract convex space is an abstract form of the classical KKM theorem. In this paper, we derive generalized forms of the Ky Fan minimax inequality, the von Neumann-Sion minimax theorem, the von Neumann-Fan intersection theorem, the Fan-type analytic alternative, and the Nash equilibrium theorem for abstract convex spaces satisfying the partial KKM principle. Th...
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ژورنال
عنوان ژورنال: The American Mathematical Monthly
سال: 2007
ISSN: 0002-9890,1930-0972
DOI: 10.1080/00029890.2007.11920401