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Displaying similar documents to “Ceva's and Menelaus' theorems for the n -dimensional space.”

A note on intersections of simplices

David A. Edwards, Ondřej F. K. Kalenda, Jiří Spurný (2011)

Bulletin de la Société Mathématique de France

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We provide a corrected proof of [1, Théorème 9] stating that any metrizable infinite-dimensional simplex is affinely homeomorphic to the intersection of a decreasing sequence of Bauer simplices.

Sperner's Lemma

Karol Pąk (2010)

Formalized Mathematics

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In this article we introduce and prove properties of simplicial complexes in real linear spaces which are necessary to formulate Sperner's lemma. The lemma states that for a function ƒ, which for an arbitrary vertex υ of the barycentric subdivision B of simplex K assigns some vertex from a face of K which contains υ, we can find a simplex S of B which satisfies ƒ(S) = K (see [10]).

Combinatorial lemmas for polyhedrons I

Adam Idzik, Konstanty Junosza-Szaniawski (2006)

Discussiones Mathematicae Graph Theory

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We formulate general boundary conditions for a labelling of vertices of a triangulation of a polyhedron by vectors to assure the existence of a balanced simplex. The condition is not for each vertex separately, but for a set of vertices of each boundary simplex. This allows us to formulate a theorem, which is more general than the Sperner lemma and theorems of Shapley; Idzik and Junosza-Szaniawski; van der Laan, Talman and Yang. A generalization of the Poincaré-Miranda theorem is also...

Brouwer Fixed Point Theorem for Simplexes

Karol Pąk (2011)

Formalized Mathematics

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In this article we prove the Brouwer fixed point theorem for an arbitrary simplex which is the convex hull of its n + 1 affinely indepedent vertices of εn. First we introduce the Lebesgue number, which for an arbitrary open cover of a compact metric space M is a positive real number so that any ball of about such radius must be completely contained in a member of the cover. Then we introduce the notion of a bounded simplicial complex and the diameter of a bounded simplicial complex....

Simplicial nonpositive curvature

Tadeusz Januszkiewicz, Jacek Świątkowski (2006)

Publications Mathématiques de l'IHÉS

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We introduce a family of conditions on a simplicial complex that we call local -largeness (≥6 is an integer). They are simply stated, combinatorial and easily checkable. One of our themes is that local 6-largeness is a good analogue of the non-positive curvature: locally 6-large spaces have many properties similar to non-positively curved ones. However, local 6-largeness neither implies nor is implied by non-positive curvature of the standard metric. One can think of these results as...

Combinatorial lemmas for polyhedrons

Adam Idzik, Konstanty Junosza-Szaniawski (2005)

Discussiones Mathematicae Graph Theory

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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.