@MISC{SIMPLEX2.MIZ,
  AUTHOR = {P\k{a}k, Karol},
  TITLE = {Brouwer Fixed Point Theorem for Simplexes},
  SECTION1 = {The Lebesgue Number},
  SECTION2 = {Bounded Simplicial Complexes},
  SECTION3 = {The Diameter of a Bounded Simplicial Complex},
  SECTION4 = {The Estimation of Diameter of the Barycentric Subdivision},
  SECTION5 = {Main Theorems},
  DAY = {21},
  MONTH = {December},
  YEAR = {2010},
  ADDRESS1 = {University of Bialystok, ul. Akademicka 2, 15-267 Bialystok},
  SUMMARY = {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 ${\cal E}^n$. First we introduce the
             Lebesgue number, which for an arbitrary open cover of a compact
             metric space $\mathfrak{M}$ is a positive real number so that
             any ball 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
	     ${\cal E}^n$. This article is based on \cite{DUDA:BM61}.}}
