Simplicial nonpositive curvature

Tadeusz Januszkiewicz; Jacek Świątkowski

Displaying similar documents to “Simplicial nonpositive curvature”

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....

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...

Ceva's and Menelaus' theorems for the $n$ -dimensional space.

Buba-Brzozowa, Malgorzata (2000)

Journal for Geometry and Graphics

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Gergonne and Nagel points for simplices in the $n$ -dimensional space.

Koźniewski, Edwin, Górska, Renata A. (2000)

Journal for Geometry and Graphics

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A non-symmetric generalization of the Borsuk -Ulam theorem

Kapil Joshi (1973)

Fundamenta Mathematicae

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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]).