I am looking for a concise and mathematically robust proof of the Sperner's Lemma.
The easiest proof I found so far is Math Pages Blog, but I don't get it without few details.
Following is the proof of the Sperner's lemma from the Math Pages Blog.
Sperner's Lemma Proof:
To prove the lemma for the $n$-th dimension all we need to do is: Let $v$ be an inner vertice in $T$. Define a linear function of $t$ that moves this vertice to the outer vertice of the same color when $t$ goes from 0 to 1.
Q:What does it mean the outer vertices, it's a vertex on of the facets of the initial simplex? What does it mean $t$ goes from 0 to 1, according to the definition it just moves any inner vertex to outer vertex.
The volume of any given simplex in $T$ is the determinant of the vectors that correspond to the vertices of the simplex.
Q:example, dimension $2$, as a result of triangulation there are triangles, vectors that correspond to vertices are 2 dimensional vectors and there three of them, how to compute the determinant in this case.
Since all the vectors are linear functions of $t$, the volume is a polynomial of degree $n$. The sum of the volumes is thus also a polynomial of degree $n$. For $t=0$ the polynomial is obviously the volume of the outer simplex (for n=2 it is the volume of ABC). However if t is only slightly bigger then zero, we still have a triangulation so the sum of the volumes is the same and therefore the polynomial is a constant. For simplicity lets say that the volume is exactly 1. Now, when t=1 the volume of all the simplexes that are not colored in n colors becomes zero. On the other hand, the volumes of the other simplexes are either 1 or (-1) (depends on orientation). And with this we are done - if the sum of the volumes is 1 there are must be simplexes that have volume 1 but this is only possible if there is an odd number of simplexes with n colors.
The rest question is due to I don't understand the nature of the function $t$ from the begging.
I tried to find this proof in more formal format, but so far with no success.
I will appreciate if someone could shed the light on this proof of the sperner's lemma. In addition, if you familiar with more intuitive proof, please share it with us.
This is a nice and intuitive proof by induction. http://math.mit.edu/~fox/MAT307-lecture03.pdf