Chapter 6

Fixed Point TheoremsThis is on fixed point theorems which feature the Brouwer fixed point theorem. This nextblock of material is a discussion of simplices and triangulations used to prove the Brouwerfixed point theorem in an elementary way. It features the famous Sperner’s lemma and isbased on very elementary concepts from linear algebra in an essential way. However, it ispretty technical stuff. This elementary proof is harder than those which come from otherapproaches like integration theory or degree theory. These other shorter ways of obtainingthe Brouwer fixed point theorem from analytical methods are presented later. If desired, thischapter could be placed after the easier to prove version of the Brouwer fixed point theorem,Theorem 11.6.8 on Page 329 after sufficient integration theory has been presented. I likethe approach presented in this chapter which is based on simplices because it is elementaryand contains a method for locating a fixed point. It seems philosophically wrong to makethis theorem depend on integration theory.

6.1 Simplices and TriangulationsDefinition 6.1.1 Define an n simplex, denoted by [x0, · · · ,xn], to be the convex hullof the n+1 points, {x0, · · · ,xn} where {xi−x0}n

i=1 are linearly independent. Thus

[x0, · · · ,xn]≡

{n

∑i=0

tixi :n

∑i=0

ti = 1, ti ≥ 0

}.

Note that{x j−xm

}j ̸=m are also independent. I will call the {ti} just described the coor-

dinates of a point x.

To see the last claim, suppose ∑ j ̸=m c j (x j−xm) = 0. Then you would have

c0 (x0−xm)+ ∑j ̸=m,0

c j (x j−xm) = 0

= c0 (x0−xm)+ ∑j ̸=m,0

c j (x j−x0)+

(∑

j ̸=m,0c j

)(x0−xm) = 0

= ∑j ̸=m,0

c j (x j−x0)+

(∑j ̸=m

c j

)(x0−xm)

Then you get ∑ j ̸=m c j = 0 and each c j = 0 for j ̸= m,0. Thus c0 = 0 also because the sumis 0 and all other c j = 0.

Since {xi−x0}ni=1 is an independent set, the ti used to specify a point in the convex hull

are uniquely determined. If two of them are ∑ni=0 tixi = ∑

ni=0 sixi.Then ∑

ni=0 ti (xi−x0) =

∑ni=0 si (xi−x0) so ti = si for i≥ 1 by independence. Since the si and ti sum to 1, it follows

that also s0 = t0. If n≤ 2, the simplex is a triangle, line segment, or point. If n≤ 3, it is atetrahedron, triangle, line segment or point.

Definition 6.1.2 If S is an n simplex. Then it is triangulated if it is the union ofsmller sub-simplices, the triangulation, such that if S1,S2 are two simplices in the triangu-lation, with

S1 ≡[z1

0, · · · ,z1m], S2 ≡

[z2

0, · · · ,z2p]

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