Sperner's lemma

The lemma of Sperner, often called Spernersches Lemma ( Sperner 's Lemma English ), is a mathematical result from the branch of topology. It goes back to the German mathematician Emanuel Sperner, who published it in 1928. The importance of the lemma is that it has proven to be a fundamental tool to prove a number of important mathematical theorems of elementary combinatorial methods, such as the brouwer between fixed-point theorem and related results, or even the set of the invariance of the open set or the patch set of Lebesgue.

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Terminology related

The following is a Euclidean space of finite dimension over the field of real numbers is based on consistent.

Simplex

If one forms in to the given affine independent points () the convex hull of these points, we obtain the n- simplex. The names of the vertices or corners of the corresponding n -simplex and its dimension. In the following, the vertices of the n -simplex is also written for the amount.

Side of a simplex

If one forms a subset of the convex hull in the same way, we obtain a sub- simplex, which is called the (r- dimensional ) page.

Simplicial complex and vertex set

A ( finite ) simplicial complex in the Euclidean space is a family of simplices of the following properties:

The sides of an n -simplex family always forms a finite simplicial complex.

By forming the union, we obtain the set of vertices of, ie the set of all vertices of the simplices in occurring.

Polyhedra and triangulation

The union, formed over all simplices of a simplicial complex, it is called to corresponding polyhedron and its triangulation. We then say the polyhedron will triangulated with. Since it is assumed here that a finite family, it always is in such a polyhedron is a compact subset of the underlying Euclidean space.

Seitpunkt and medium point

A point is called a Seitpunkt of when in a real page is included ( with ). Otherwise, it is called the mean point of.

Is thus a central point if and only if its formed with respect to the vertices of barycentric coordinates are all larger. Accordingly, if and only one of Seitpunkt when one of its barycentric coordinates with respect formed is the same.

Carrier simplex

Always exists exactly one side of which is a central point for a point. It is the side of smallest dimension among all the pages in which is included. This is called short, the support simplex of ( in ).

The corresponding to the corners of this page index set is referred to below.

Spernersche Eckpunktbezifferung and complete simplices

Is an n- simplex fixed and to a ( finite ) simplicial complex which triangulates this simplex, and is more a figure which satisfies the condition for each corner ( Sperner condition), so one calls such as Eckpunktbezifferung or Spernersche Eckpunktbezifferung (English Sperner labeling ).

For each simplex is then reacted.

It is obviously always. Applies even, as one calls such a simplex as complete.

Formulation of the lemma Spernerschen

The Spernersche lemma can be formulated as follows: