Equivalence Between An Approximate Version Of Brouwer s Fixed Point Theorem And Sperner s Lemma: A Constructive Analysis

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1 Applied Mathematics E-Notes, 11(2011), c ISSN Available free at mirror sites of Equivalece Betwee A Approximate Versio Of Brouwer s Fixed Poit Theorem Ad Sperer s Lemma: A Costructive Aalysis Yasuhito Taaka y Received 8 November 2010 Abstract It is widely believed that Sperer s lemma ad Brouwer s xed poit theorem are equivalet. But i secod order arithmetic ([5]), although Sperer s lemma is proved i RCA 0, Brouwer s xed poit theorem is ot. Also i Bishop style costructive mathematics, although Sperer s lemma ca be costructively proved, Brouwer s xed poit theorem ca ot be costructively proved. We cosider a approximate (or a costructive) versio of Brouwer s xed poit theorem, ad show the equivalece betwee Sperer s lemma ad a approximate versio of Brouwer s xed poit theorem. We follow the Bishop style costructive mathematics accordig to [1], [2] ad [3]. 1 Itroductio Brouwer s xed poit theorem is extesively applied i ecoomic theory ad game theory. It is widely believed that Sperer s lemma ad Brouwer s xed poit theorem are equivalet. But i secod order arithmetic ([5]), although Sperer s lemma is proved i RCA 0, Brouwer s xed poit theorem is ot. Also i Bishop style costructive mathematics, although Sperer s lemma ca be costructively proved, Brouwer s xed poit theorem ca ot be costructively proved. Recetly some authors have preseted a approximate (or a costructive) versio of Brouwer s xed poit theorem usig Sperer s lemma. See [4] ad [8]. We will show that Sperer s lemma ad a approximate versio of Brouwer s xed poit theorem are equivalet. We follow the Bishop style costructive mathematics accordig to [1], [2] ad [3]. 2 Approximate Versio of Brouwer s Theorem Let p = (p 0 ; p 1 ; : : : ; p ) be a poit i a -dimesioal simplex, ad cosider a fuctio ' from to itself. is a ite atural umber. We de e uiform cotiuity ad approximate xed poit as follows. Mathematics Subject Classi catios: 03F65, 26E40. y Faculty of Ecoomics, Doshisha Uiversity, Kamigyo-ku, Kyoto, , Japa. 238

2 Y. Taaka 239 Uiform cotiuity of fuctios A fuctio ' is uiformly cotiuous if for ay p, p 0 ad " > 0 there exists > 0 such that Approximate xed poit xed poit of ' if If jp 0 pj < ; the j'(p 0 ) '(p)j < ": For each " p is a approximate (or a "-approximate) jp '(p)j < ": A approximate versio of Brouwer s xed poit theorem is as follows. THEOREM 1. (Approximate versio of Brouwer s xed poit theorem) For each " > 0 ay uiformly cotiuous fuctio from a -dimesioal simplex to itself has a approximate xed poit. PROOF. See [4] or Theorem 6 i [8] 1. 3 From Approximate Versio of Brouwer s Theorem to Sperer s Lemma Let us partitio the simplex. Figure 1 is a example of partitio (triagulatio) of a 2-dimesioal simplex. I a 2-dimesioal case we divide each side of i m equal segmets, ad draw the lies parallel to the sides of. The, the 2-dimesioal simplex is partitioed ito m 2 triagles. We cosider partitio of iductively for cases of higher dimesio. I a 3 dimesioal case each face of is a 2-dimesioal simplex, ad so it is partitioed ito m 2 triagles i the above metioed way, ad draw the plaes parallel to the faces of. The, the 3-dimesioal simplex is partitioed ito m 3 trigoal pyramids. Ad similarly for cases of higher dimesio. Let K deote the set of small -dimesioal simplices of costructed by partitio. Vertices of these small simplices of K are labeled with the umbers 0, 1, 2, : : :, subject to the followig rules. 1. The vertices of are respectively labeled with 0 to. We label a poit (1; 0; : : : ; 0) with 0, a poit (0; 1; 0; : : : ; 0) with 1, a poit (0; 0; 1 : : : ; 0) with 2, : : :, a poit (0; : : : ; 0; 1) with. That is, a vertex whose k-th coordiate (k = 0; 1; : : : ; ) is 1 ad all other coordiates are 0 is labeled with k. 2. If a vertex of K is cotaied i a 1-dimesioal face of, the this vertex is labeled with some umber which is the same as the umber of a vertex of that face. 3. If a vertex of K is cotaied i a 2-dimesioal face of, the this vertex is labeled with some umber which is the same as the umber of a vertex of that face. Ad similarly for cases of lower dimesio. 1 [4] ad [8] have show the theorem for oly a 2-dimesioal case. But it ca be exteded to a geeral -dimesioal case. I [6] ad [7] we have costructively show a approximate versio of Brouwer s xed poit theorem for a -dimesioal simplex ad other xed poit theorems.

3 240 Equivalece betwee Brower s Theorem ad Sperer s Lemma Figure 1: Partitio ad labelig of 2-dimesioal simplex 4. A vertex cotaied i iside of is labeled with a arbitrary umber amog 0, 1, : : :,. A small simplex of K which is labeled with the umbers 0, 1, : : :, is called a fully labeled simplex. Sperer s lemma is as follows; LEMMA 1. (Sperer s lemma) If we label the vertices of K followig above rules 1 4, the there exists at least oe fully labeled simplex. Now we show the followig result. THEOREM 2. Sperer s lemma is derived from the approximate versio of Brouwer s xed poit theorem. PROOF. Deote vertices of a -dimesioal simplex of K by x 0 ; x 1 ; : : : ; x, ad deote the j-th compoet of x i by x i j. These vertices are labeled accordig to the above rules 1 4. Deote the label of x i by l(x i ). Let be a positive umber which is smaller tha x i l(x i ) for all xi, ad de e a fuctio f(x i ) as follows 2 ; ad f(x i ) = (f 0 (x i ); f 1 (x i ); : : : ; f (x i )); f j (x i ) = x i j for j = l(x i ); x i j + for j 6= l(x i ): f j deotes the j-th compoet of f. From the labelig rules x i l(x i ) > 0 for all xi, ad so > 0 is well de ed. Sice P j=0 f j(x i ) = P j=0 xi j = 1, we have f(x i ) 2 : 2 We refer to [9] about the de itio of this fuctio. (1)

4 Y. Taaka 241 We exted f to all poits i the simplex by covex combiatios of its values o the vertices of the simplex. Let z be a poit i the -dimesioal simplex of K whose vertices are x 0 ; x 1 ; : : : ; x. The, z ad f(z) are represeted as follows; z = i x i ; ad f(z) = i f(x i ); i 0; i = 1: Let us show that f is uiformly cotiuous. Let z ad z 0 be distict poits i the same small -dimesioal simplex of K. They are represeted as ad so The, we have ad for each j z z 0 = z = i x i ; z 0 = ( i 0 i)x i ad z j zj 0 = f(z) f(z 0 ) = i:j6=l(i) 0 ix i ; ( i 0 i)x i j for each j: ( i 0 i)f(x i ) f j (z) f j (z 0 ) = ( i 0 i)x i j + X ( i 0 i) X ( i 0 i) i:j6=l(i) i:j=l(i) = z j zj 0 + X ( i 0 i) X ( i 0 i) i:j=l(i) Sice is ite, appropriately selectig 0 i give i for each i we ca make jf j (z) f j (z 0 )j su cietly small correspodig to the value of jz j zj 0 j for each j, ad so make jf(z) f(z 0 )j su cietly small correspodig to the value of jz z 0 j. Thus, f is uiformly cotiuous, ad the by the approximate versio of Brouwer s xed poit theorem there exists a poit z such that for ay " > 0. The, we obtai jz f(z )j < " jz i f i (z )j < " for all i: Let > 0 ad ~z be a poit i V (z ; ), where V (z ; ) is a -eighborhood of z. If is su cietly small, uiform cotiuity of f meas j~z i f i (~z)j < " (2) for ay " > 0 ad for all i. ~z i is the i-th compoet of ~z. Let be a simplex of K which cotais ~z.

5 242 Equivalece betwee Brower s Theorem ad Sperer s Lemma We caot costructively determie which small simplex i K cotais z. But we ca costructively determie which small simplex has a itersectio with V (z ; ), or we may cosider that if is su cietly small, all poits i V (z ; ) are approximate xed poits. Let z 0 ; z 1 ; : : : ; z be the vertices of. The, ~z ad f(~z) are represeted as ~z = i z i ad f(~z) = i f(z i ); i 0; i = 1: (1) implies that if oly oe z k amog z 0 ; z 1 ; : : : ; z is labeled with i, we have 0 1 jf i (~z) ~z i j = j z j i + X j k zi 1 j k A < ": j=0 j=0;j6=k Sice " may be arbitrarily small ad > 0, this meas 1 j=0;j6=k j k 0: j=0;j6=k (2) is satis ed with k 1 +1 for all k. O the other had if o zj is labeled with i, we have f i (~z) = j z j i = ~z i + (1 + 1 ); j=0 ad the (2) ca ot be satis ed. Thus, for each i oe ad oly oe z j must be labeled with i. Therefore, must be a fully labeled simplex. We have completed the proof of Sperer s lemma by the approximate versio of Brouwer s xed poit theorem. Sperer s lemma aloe is ot su ciet to prove Brouwer s xed poit theorem. We eed some o-costructive argumets. But Sperer s lemma is su ciet to costructively prove a approximate versio of Brouwer s xed poit theorem. Ad coversely a approximate versio of Brouwer s xed poit theorem is su ciet to prove Sperer s lemma. Ackowledgmet. I thak the aoymous referee for costructive remarks ad suggestios that greatly improved the origial mauscript of this paper. This research was partially supported by the Miistry of Educatio, Sciece, Sports ad Culture of Japa, Grat-i-Aid for Scieti c Research (C), Refereces [1] E. Bishop ad D. Bridges, Costructive Aalysis, Spriger, [2] D. Bridges ad F. Richma, Varieties of Costructive Mathematics, Cambridge Uiversity Press, 1987.

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