Brouwer Fixed Point Theorem for Simplexes

Karol Pąk

Formalized Mathematics (2011)

  • Volume: 19, Issue: 3, page 145-150
  • ISSN: 1426-2630

Abstract

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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. 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].

How to cite

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Karol Pąk. "Brouwer Fixed Point Theorem for Simplexes." Formalized Mathematics 19.3 (2011): 145-150. <http://eudml.org/doc/267902>.

@article{KarolPąk2011,
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].},
author = {Karol Pąk},
journal = {Formalized Mathematics},
language = {eng},
number = {3},
pages = {145-150},
title = {Brouwer Fixed Point Theorem for Simplexes},
url = {http://eudml.org/doc/267902},
volume = {19},
year = {2011},
}

TY - JOUR
AU - Karol Pąk
TI - Brouwer Fixed Point Theorem for Simplexes
JO - Formalized Mathematics
PY - 2011
VL - 19
IS - 3
SP - 145
EP - 150
AB - 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].
LA - eng
UR - http://eudml.org/doc/267902
ER -

References

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