<abstract xml:lang="en" xmlns="http://www.w3.org/1999/xhtml">In this article we prove the Brouwer fixed point theorem for an arbitrary simplex which is the convex hull of its <italic>n</italic> + 1 affinely indepedent vertices of <italic>εn</italic>. 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. We also prove the estimation of diameter decrease which is connected with the barycentric subdivision. Finally, we prove the Brouwer fixed point theorem and compute the small inductive dimension of <italic>εn</italic>. This article is based on [16].</abstract>

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. We also prove the estimation of diameter decrease which is connected with the barycentric subdivision. Finally, we prove the Brouwer fixed point theorem and compute the small inductive dimension of εn. This article is based on [16].

Brouwer Fixed Point Theorem for Simplexes

Institute of Informatics, University of Białystok, Poland

Formalized Mathematics

{"article-title":"Brouwer Fixed Point Theorem for Simplexes"}

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....

Brouwer Fixed Point Theorem for Simplexes

Publicado en línea: 26 abr 2012

Páginas: 145 - 150

DOI: https://doi.org/10.2478/v10037-011-0023-4

This content is open access.