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Course: M393C: equilibrium theory Term: Spring 2007 Instructor: Gordan Zitkovic
Lecture 3
Two Fixed-Point Theorems
This lecture is based on selected parts of [Bor03].
3.1. Sperner’s Lemma
Sperner’s lemma is due to Sperner (see [Spe29]). The proof given here is due to H.W. Kuhn (see [Kuh68]).
Let ∆ =
x ∈ Rn + :
∑n i=1 xi^ = 1
stand for the unit simplex in Rn. The points in ∆ the Euclidean coordinates on ∆ are also called the barycentric coordinates. For m ∈ N, let Sm denote the set of all x ∈ ∆ such that mxi ∈ N ∪ { 0 }, for all i = 1,... , n, where x = (x 1 ,... , xn). The points in Sm divide the simplex ∆ into mn−^1 subsimplices, the set of which will be denoted by σm. Also, each (n − 1)-dimensional face of ∆ admits a similar decomposition into mn−^2 (n − 1)-dimensional simplices. Let us denote the set of all these by ∂σ m′. Clearly, each δ′^ ∈ ∂σ m′ is an (n − 1)-dimensional face of a unique δ ∈ σm. The set of all (n − 1)-dimensional faces of all δ ∈ σm (some of which will not be on the bounday of ∆) will be denoted by ∂σm. Thus, ∂σ′ m is a subset of ∂σm consisting of those (n − 1)-dimensional faces that lie on the boundary of ∆. A labeling is a function λ : Sm → { 1 , 2 ,... , n}. A labeling is called proper if λ(x) ∈ P (x), for all x ∈ Sm, where P (x) = {i = 1,... , n : xi 6 = 0}. A subsimplex δ ∈ σm is said to be completely labeled if λ(δ) = { 1 , 2 ,... , n}, where λ(δ) = {λ(x) : x ∈ δ}).
Theorem 3.1 (Sperner’s Lemma). Suppose that λ is a proper labeling on Sm. Then there exists an odd number of completely labeled subsimplex δ ∈ σm.
Proof. Induction on the dimension n ∈ N of the simplex ∆. When n = 1, the statement is trivially true. If n = 2, when ∆ is a line segment, the claim is clear. Suppose that is holds for all k < n, for some n > 1, and let us prove that it also holds for n. Pick a proper labeling λ, and define the following sets,
C = {δ ∈ σm : δ is completely labeled } , A = {δ ∈ σm : λ(δ) = { 1 , 2 ,... , n − 1 }} , B = {δ ∈ ∂σ′ m : δ which bear all labels in the set { 1 , 2 ,... , n − 1 }} , E = {δ ∈ ∂σm : δ which bear all labels in the set { 1 , 2 ,... , n − 1 }}
The goal is, of course, to show that |C| is odd. We start by constructing a graph Γ in the following manner (draw a picture for the case n = 3): the set of nodes (vertices) of Γ is A ∪ B ∪ C, and the set of edges is E. Clearly, there are only two possibilities for each δ ∈ E: (a) is either a common face of two subsimplices, in which case it connects two nodes in A ∪ C, or, (b) it is a boundary face, and then it connects its carrying subsimplex in A ∪ C with δ ∈ B. A degree of a vertex δ ∈ A ∪ B ∪ C is defined as the number of edges it is an end-point of. Recall that the number of vertices with odd degrees is necessarily even (why?). The degree of the vertices in A is always 2, and the degree of those in C is always 1. For those in B, the degree is also always 1, as they are connected to unique subsimplices thay are faces of. Therefore, the parity of the number of vertices in C matches the parity of the number of vertices in B. However, the number of vertices in B is odd, by the induction hypothesis, and, therefore, so is the number of vertices in C.
Corollary 3.2. There exists at least one completely labeled subsimplex for any proper labeling λ.
3.2. Brouwer’s fixed-point theorem
Brouwer’s fixed-point theorem is, of course, due to Brouwer (see [Bro12]).
As before, let ∆ denote the unit simplex in Rn.
Theorem 3.3. Any continuous function f : ∆ → ∆, has a fixed point.
Proof. For m ∈ N, let the subdivision Sm be defined as above. For each x ∈ Sm, let (f 1 (x), f 2 (x),... , fn(x)) be the (barycentric) coordinates of f (x), and let (x 1 ,... , xn) be the coordinates of x. Since
∑n ∑n i=1^ fi(x) = i=1 x^1 = 1, for each^ x^ ∈^ Sm^ there exists some index^ λ(x) such that^ fλ(x)(x)^ ≤^ xλ(x). Furthermore,^ λ(x) can (and will) always be chosen among those indices i such that xi 6 = 0 (why?). The mapping λ : Sm → { 1 , 2 ,... , n} is a labeling in the sense of the previous section. Therefore, by Corollary 3.2 (Sperner’s Lemma) for each m ∈ N, there exists a fully labeled subsimplex δm of ∆. Let the full set of vertices of δk be denoted by
xk,^1 ,... , xk,n
, where we choose the numbers so that the label λ(xk,j^ )
of xk,j^ is j. We know that yik,i ≤ xk,ii , where yk,i^ = f (xk,i) ∈ ∆, i ∈ { 1 , 2 ,... , n}. Thanks to compactness of ∆, a convergent subsequence {xk(l),^1 }l∈N with limit x∞^ ∈ ∆ can be extracted from {xk,^1 }k∈N, Continuity of f and the fact that the diameter of δk shrinks to 0 imply that xk(l),j^ → x∞^ and yk(l),j^ → y∞^ = f (x∞), for all j ∈ { 1 , 2 ,... , n}, as l → ∞. By the defining property of the labeling λ, the following holds for all i, j ∈ { 1 , 2 ,... , n}
x∞ i = lim n xk,ii ≤ lim n yk,ii = y∞ i.
Both x∞^ and y∞^ are (barycentric coordinates of) points in ∆, so
∑n i=1 x
∞ i =^
∑n i=1 y
∞ i = 1. It follows that x∞^ = y∞^ = f (x∞), so x∞^ is a fixed point of f.
Let A, B be two subsets of Rn. A map f : A → B is called a homeomorphism if it is continuous, invertible and its inverse f −^1 : B → A is continuous as well. If there is a homeomorphism from A to B, then the sets A and B are said to be homeomorphic.
Proposition 3.4. Let A be a subset of Rn^ homeomorphic to the unit simplex ∆. Then each continuous function g : A → A has a fixed point.
Proof. Let f be a homeomorphism from A to ∆. Define the mapping h : ∆ → ∆ by h(x) = f (g(f −^1 (x))). Clearly, h is a continuous function mapping ∆ into itself, so it admits a fixed point x. Then, with y = f −^1 (x) has the property that g(y) = g(f −^1 (x)) = f −^1 (h(x)) = f −^1 (x) = y.
Example 3.5. Any convex, closed and bounded subset of Rn^ is homeomorphic to ∆ (see Problem 3.2).
3.3. A theorem of Knaster, Kuratowski and Mazurkiewicz
First proved in [KKM29], this theorem is, perhaps, even more useful than Brower’s. Truth be told, it is not exactly a fixed-point theorem, but it definitely feels like one.
Theorem 3.6 (Knaster, Kuratowski and Mazurkiewicz). Let F 1 , F 2 ,... , Fn be closed subsets of the unit simplex ∆ ∈ Rn. Let x^1 ,... , xn^ be the vertices of ∆, and suppose that for any I ⊆ { 1 , 2 ,... , n} we have
conv
xi^ : i ∈ I
i∈I
Fi
where conv B denotes the convex hull of B, i.e., the smallest convex set containing B. Then
⋂^ n
i=
Fi 6 = ∅.
[KKM29] B. Knaster, C. Kuratowski, and S. Mazurkiewicz, Ein beweis des fixpunktsatsez f¨ur n-dimensionale simplexe, Fund. Math. (1929), no. XIV, 132–137. [Kuh68] Harold W. Kuhn, Simplicial approximation of fixed points, Proc. Nat. Acad. Sci. U.S.A. 61 (1968), 1238–1242. [Spe29] E. Sperner, Neuer Beweis f¨ue die Invarianz der Dimansionszahl und des Gebietes, Abhandlungen aus dem Mathe- matischen Seminar der Hamburgischen Universit¨at 6 (1929), 265–272.
