Sperner's Lemma

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, Sperner's lemma is a combinatorial analog of the Brouwer fixed point theorem. Sperner's lemma states that every Sperner coloring of a triangulation of an n-dimensional simplex contains a cell colored with a complete set of colors. The initial result of this kind was proved by Emanuel Sperner, in relation with proofs of invariance of domain. Sperner colorings have been used for effective computation of fixed points, in root-finding algorithms, and are applied in fair division algorithms. According to the Soviet Mathematical Encyclopaedia (ed. I.M. Vinogradov), a related 1929 theorem (of Knaster, Borsuk and Mazurkiewicz) has also become known as the Sperner lemma - this point is discussed in the English translation (ed. M. Hazewinkel). It is now commonly known as the Knaster-Kuratowski-Mazurkiewicz lemma.