proof of Brouwer fixed point theorem

The n-dimensional simplex 𝒮n is the following subset of ℝn+1

{(α1,α2,…,αn+1)|∑i=1n+1αi=1,αi≥0 ∀i=1,…,n+1}

Given an element x=∑iαi⁢ei∈𝒮n we denote [x]i=αi (i.e., the i-th barycentric coordinateMathworldPlanetmath). We also denote F⁢(x)={i|[x]i≠0}. An I-face of 𝒮n is the subset {x|F⁢(x)⊆I}.

As was noted in the statement of the theorem, the ’shape’ is unimportant. Therefore, we will prove the following variant of the theorem using the KKM lemma.

Theorem 1 (Brouwer’s Fixed Point Theorem).

Let f:Sn→Sn be a continuous functionMathworldPlanetmathPlanetmath. Then, f has a fixed pointPlanetmathPlanetmath, namely, there is an L∈Sn such that L=f⁢(L).


Clearly, ∑i=1n[y]i=1 for any y∈𝒮n and L=f⁢(L) if and only if [L]i=[f⁢(L)]i for all i=1,2,…,n+1. For each i=1,2,…,n+1 we define the following subset Ci of 𝒮n:


We claim that if x is in some I-face of 𝒮n (I⊆{1,2,…,n+1}) then there is an index i∈I such that x∈Ci. Indeed, if x is in some I-face then F⁢(v)⊆I. Thus, if [x]i≠0 then i∈I. This shows that


Assuming by contradictionMathworldPlanetmathPlanetmath that x∉Ci for all i∈I implies that [x]i<[f⁢(x)]i for all i∈I. But this leads to a contradiction as the following inequalityMathworldPlanetmath shows:


This dicussion establishes that each I-face is contained in the union ∪i∈ICi. In addition, the subsets Ci are all closed. Therefore, we have shown that the hypothesisMathworldPlanetmath of the KKM Lemma holds.

By the KKM lemma there is a point L that is in every Ci for i=1,2,…,n+1. We claim that L is a fixed point of f. Indeed, [L]i≥[f⁢(L)]i≥0 for all i=1,2,…,n+1 and thus:


Therefore, [L]i=[f⁢(L)]i for all i=1,2,…,n+1 which implies that L=f⁢(L). ∎

Title proof of Brouwer fixed point theoremPlanetmathPlanetmath
Canonical name ProofOfBrouwerFixedPointTheorem1
Date of creation 2013-03-22 18:13:24
Last modified on 2013-03-22 18:13:24
Owner uriw (288)
Last modified by uriw (288)
Numerical id 4
Author uriw (288)
Entry type Proof
Classification msc 47H10
Classification msc 54H25
Classification msc 55M20