Sperner's Lemma states that for a triangulation of a simplex whose vertices are colored according to a specific boundary rule, at least one of the small triangulated cells must have all of its own vertices colored with the full set of colors. Named for Emanuel Sperner, it is a purely combinatorial statement that nonetheless supplies one of the simplest known proofs of the Brouwer Fixed-Point Theorem, since the fully colored cell it guarantees can be used to locate a fixed point as the triangulation is refined.
Facts
StatementEvery Sperner coloring of a triangulation of an n-dimensional simplex contains a cell whose vertices all have different colors. 1 Classification
Statement Form Connections
Sources
1. Sperner's lemma (Wikipedia)
StatementQuote, Statement
every Sperner coloring of a triangulation of an n-dimensional simplex contains a cell whose vertices all have different colors.
Reader Challenges (0)
No disputes yet. Spotted an error or a better source? Open the first one.
Sign in to dispute this or suggest a correction.