any point of , if it is on the side , satisfies
similarly if is on the side , then
if is on the side , then
(It follows that .) Then some (triangular) simplex of , say , satisfies
We will informally sketch a proof of a stronger statement: Let (resp. ) be the number of simplexes satisfying (1) and whose vertices have the same orientation as (resp. the opposite orientation). Then (whence ).
Let’s define a “circuit” of size as an injective mapping of the cyclic group into such that is adjacent to for all in the group.
Any circuit has what we will call a contour integral , namely
Let us say that two vertices and are equivalent if .
There are four steps:
1) Contour integrals are added when their corresponding circuits are juxtaposed.
2) A circuit of size 3, hitting the vertices of a simplex , has contour integral
0 if any two of , , are equivalent, else
+3 if they are inequivalent and have the same orientation as , else
3) If is a circuit which travels around the perimeter of the whole triangle , and with same orientation as , then .
4) Combining the above results, we get
where the sum contains one summand for each simplex .
Remarks: In the figure, and : there are two “red-green-blue” simplexes and one blue-green-red.
With the same hypotheses as in Sperner’s lemma, there is such a simplex which is connected (along edges of the triangulation) to the side (resp. ,) by a set of vertices for which (resp. , . The figure illustrates that result: one of the red-green-blue simplexes is connected to the red-green side by a red-green “curve”, and to the other two sides likewise.
The original use of Sperner’s lemma was in a proof of Brouwer’s fixed point theorem in two dimensions.
|Date of creation||2013-03-22 13:44:33|
|Last modified on||2013-03-22 13:44:33|
|Last modified by||mathcam (2727)|