# Labelling rules and orientation: On Sperner's lemma and brouwer degree

• G. V. D. Laan
• A. J. J. Talman
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 878)

## Abstract

In this paper we consider two labelling rules used in simplicial fixed point algorithms. The first one is the standard labelling rule from an n-dimensional set to the set of integers {1, …, n+1}. The second one is a labelling to the set {±i|i = 1, …, n}. The main purpose of the paper is to compare the two rules. We define the orientation of a completely labelled simplex and give some generalizations of the lemma of Sperner and the related lemma of Knaster, Kuratowski and Mazurkiewicz. Also, for both labelling rules it is shown that the Brouwer degree can be obtained from the completely labelled simplices.

## Keywords

Sign Vector Boundary Property Standard Labelling Unit Simplex Fixed Point Algorithm
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