Fixed-point-like theorems on subspaces

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Fixed-point-like theorems on subspaces

Philippe Bich, Bernard Cornet

To cite this version:

Philippe Bich, Bernard Cornet. Fixed-point-like theorems on subspaces. Fixed Point Theory and Applications, SpringerOpen, 2004, 2004, pp.602917 - 602917. �10.1155/S1687182004406056�. �halshs-01394846�

FIXED-POINT-LIKE THEOREMS ON SUBSPACES

Received 8 June 2004

We prove a fixed-point-like theorem for multivalued mappings defined on the finite Cartesian product of Grassmannian manifolds and convex sets. Our result generalizes two different kinds of theorems: the fixed-point-like theorem by Hirsch et al. (1990) or Husseini et al. (1990) and the fixed-point theorem by Gale and Mas-Colell (1975) (which generalizes Kakutani’s theorem (1941)).

1. Introduction

In this paper, we prove a fixed-point-like theorem for multivalued mappings defined on the finite Cartesian product of Grassmannian manifolds and convex sets. Letk be an in-teger and letV be a Euclidean space such that 0≤k≤dimV, then the k-Grassmannian manifold ofV, denoted Gk(V), is the set of all the k-dimensional subspaces of V. The setGk(V) is a smooth compact manifold but, in general, it does not satisfy properties such as convexity or acyclicity and its Euler characteristic may be null. This prevents the use of classical fixed-point theorems as Brouwer’s [2], Kakutani’s [14], or Eilenberg-Montgomery’s theorem [7].

Our main result generalizes two different kinds of theorems: the fixed-point-like the-orem by Hirsch et al. [11] or Husseini et al. [13] and the fixed-point theorem by Gale and Mas-Colell [8] (which generalizes Kakutani’s theorem [14]). As in [11,13], we will mainly use techniques from degree theory. As a consequence of our main result, we first deduce the standard fixed-point theorems when the variable is in a convex domain (such as Brouwer and Kakutani’s theorem) and second Borsuk-Ulam’s theorem.

The main result of this paper is directly motivated by the existence problem of equilib-ria in economic models with incomplete markets; in [1], it is used to extend the classical existence result by Duffie and Shafer [6] to the nontransitive case.

The paper is organized as follows. The main result is stated inSection 2together with some direct consequences of it, namely, the results by Hirsch et al. [11], Gale and Mas-Colell [8] and Borsuk-Ulam’s theorem. The proof of the main result is given inSection 3

Copyright©2004 Hindawi Publishing Corporation Fixed Point Theory and Applications 2004:3 (2004) 159–171 2000 Mathematics Subject Classification: 47H04, 47H10, 47H11 URL:http://dx.doi.org/10.1155/S1687182004406056

and the appendix recalls the main properties of the Grassmannian manifold, used in this paper.

2. Statement of the results

2.1. Preliminaries. A correspondenceΦ from a set X to a set Y is a map from X to the set

of all the subsets ofY, and the graph of Φ, denoted G(Φ), is defined by G(Φ)= {(x, y)∈

X×Y|y∈Φ(x)}. A mappingϕ : X→Y is said to be a selection of Φ if ϕ(x)∈Φ(x) for allx∈X. If A is a subset of X, we let Φ(A)=
x∈AΦ(x), and the restriction of Φ to A,

denotedΦ|A, is the correspondence fromA to Y defined by Φ|A(x)=Φ(x) if x∈A. If X

andY are topological spaces, the correspondence Φ is said to be lower semicontinuous (l.s.c.) (resp., upper semicontinuous (u.s.c.)) if for every open setU⊂Y, the set{x∈X|

Φ(x)∩U= ∅}is open inX (resp., the set{x∈X|Φ(x)⊂U}is open inX and, for every x∈X, Φ(x) is compact).

Ifx=(x1,...,xn) andy=(y1,..., yn) belong toRn, we denote byx·y=ni=1xiyithe scalar product of Rn, x =√x·x the Euclidian norm. If x∈Rn _{and r∈R}

+, we let B(x,r)= {y∈Rn_{| x−y< r}and B(x,r)= {y∈R}n_{| x−y ≤r}. IfE is a vector}

subspace ofRn, we denote byE⊥= {u∈Rn_{| ∀x∈E, x·u=0}the orthogonal space}

toE. If u1,...,uk belong toE, a vector space, we denote by span{u1,...,uk}the vector subspace ofE spanned by u1,...,uk.

LetV be a Euclidean space and let k be an integer such that 0≤k≤dimV; we denote byGk(V) the set consisting of all the linear subspaces of V of dimension k, called the (k-)Grassmannian manifold of V. Then it is known that Gk_{(V) is a smooth manifold}

of dimensionk(dimV−k) and we refer to the appendix for the properties we will use hereafter, together with the precise definition of the manifold structure onGk(V).

2.2. The main result and some consequences. The aim of this paper is to prove the

following result.

Theorem 2.1. LetI, J be two finite disjoint sets. For every i∈I, let kibe an integer and let Vi be a Euclidean space such that 0≤ki≤dimVi. For every j∈J, let Cj be a nonempty,

convex, compact subset of a Euclidean spaceVj, and letM=i∈IGki(Vi)×j∈JCj.

Fori∈I and k=1,...,ki, letFikbe a correspondence fromM to Viwith convex values,

for j∈J, let Fjbe a correspondence fromM to Cjwith convex values, and suppose that, for

everyi∈I and k=1,...,ki(resp., j∈J), the correspondence Fik(resp.,Fj) is either l.s.c or

u.s.c.

Then, there exists ¯x=(( ¯xi)i∈I, ( ¯xj)j∈J)∈M such that

(i) eitherFik( ¯x)∩x¯i= ∅orFik( ¯x)= ∅for everyi∈I and k=1,...,ki; (ii) eitherFj( ¯x)∩ {x¯j} = ∅orFj( ¯x)= ∅for everyj∈J.

The proof ofTheorem 2.1is given inSection 3. A first consequence ofTheorem 2.1is the following theorem by Hirsch et al. [11].

Corollary 2.2. LetV1be a Euclidean space, letk1be an integer such that 0≤k1≤dimV1, and for everyk=1,...,k1, letfk:Gk1(V1)→V1be a continuous mapping. Then, there exists

¯

x∈Gk1(V

Proof. TakeI= {1},J= ∅, andF1k(x)= {fk(x)}for everyx∈Gk1(V1) and for everyk= 1,...,k1. FromTheorem 2.1, there exists ¯x∈M=Gk1(V1) such that for everyk=1,...,k1, F1k( ¯x)∩x¯= ∅, that is, fk( ¯x)∈x.¯
A second consequence ofTheorem 2.1is the following generalization of Gale and Mas-Colell’s theorem [8], which is also a generalization of Kakutani’s theorem. Hereafter, we use the formulation by Gourdel [9] allowing each correspondence to be either u.s.c. or l.s.c.

Corollary 2.3. LetJ be a finite set, for j∈J, let Cjbe a nonempty, convex, compact subset of a Euclidean space, and letFjbe a correspondence fromM :=j∈JCjtoCjwith convex

values, such that the correspondenceFjis either l.s.c or u.s.c. Then, there exists ¯x=( ¯xj)j∈J∈

M such that for every j∈J, either ¯xj∈Fj( ¯x) or Fj( ¯x)= ∅.

Proof. TakeI= ∅and applyTheorem 2.1.

Remark 2.4. According to our definition, an u.s.c. correspondence has compact values and without this requirement,Theorem 2.1may not be true, as we can see in the fol-lowing counterexample. LetM :=G1_(R2_{). Each elementD of G}1_(R2_{) can be written as} Dt= {λ(cost,sint)|λ∈R}, for somet∈[0,π[. We define the correspondence F from M toR2 _byF(D

0)=R× {1}andF(Dt)=Dt∩(R× {1}) +{(1, 0)}ift∈]0,π[. We let the reader check that for every open setU⊂R2_{, the set{x∈M|F(x)⊂U}is open in} M and that F has nonempty, convex (and closed) values. Yet, it is straightforward that F(x)∩x= ∅for everyx∈G1_(R2_).

Another consequence of our main result is the following multivalued version of Borsuk and Ulam’s theorem. We denote bySnthe unit sphere ofRn+1.

Corollary 2.5. Fork=1,...,n, let Fkbe a correspondence fromSntoRwith nonempty and convex values such that for everyk=1,...,n, Fkis either l.s.c or u.s.c. Then, there exists

¯

x∈Sn_{such that}

∀k∈ {1,...,n}, Fk( ¯x)∩Fk_{(−x)¯ = ∅. (2.1)}

Proof. For everyk=1,...,n, let ˆFkbe the correspondence fromSntoRdefined by ˆ

Fk_(x)=_u−v|u∈Fk_{(x), v∈F}k_(−x)_{. (2.2)}

We let the reader check that for everyk=1,...,n, the correspondence ˆFk has non-empty, convex values and that it is u.s.c. (resp., l.s.c.) ifFk is u.s.c. (resp., l.s.c.). So, to prove Corollary 2.5, it suffices to show the existence of ¯x∈Sn_{such that 0∈Fˆ}k_{( ¯x) for}

everyk=1,...,n.

We define, for everyk=1,...,n, the correspondence Hk fromGn(Rn+1) to Rn+1 as follows: for everyE∈Gn_(Rn+1_{), we letH}k_(E)=Fˆk_{(x)x, where x is an arbitrary element of}

E⊥_∩Sn_{. The correspondenceH}k_{is well defined sinceE}⊥_∩Sn_{= {x,−x}for some element}

TakeI= {1},V1=Rn+1,k1=n, J= ∅, and applyTheorem 2.1to the correspondences Hk_{, which clearly satisfy the assumptions ofTheorem 2.1. So there exists ¯E∈G}n_(Rn+1₎

such that ¯E∩Hk_{( ¯E)= ∅for everyk=1,...,n.}

Now, if ¯x is an arbitrary point of ¯E⊥∩Sn_{, then we have ¯E∩Fˆ}k_{( ¯x)¯x= ∅; from ¯x∈E¯}⊥

and ¯x=0, we finally obtain 0∈_Fˆk_{( ¯x) for every k=1,...,n, which ends the proof of}

Corollary 2.5.

3. Proof ofTheorem 2.1

The proof is given in three steps, corresponding to the following three subsections. The first step gives the proof under the additional assumptions thatJ= ∅and the correspon-dencesFikare single-valued. The second step only assumes in addition thatJ= ∅. Finally,

the third step gives the proof under the assumptions ofTheorem 2.1.

3.1. Proof whenJ= ∅andFikare single-valued. We first proveTheorem 2.1under the

additional assumptions thatJ= ∅and theFikare single-valued. This is exactly the

state-ment below.

Theorem 3.1. LetI be a finite set and for i∈I, let kibe an integer and letVibe a Euclidean

space such that 0≤ki≤dimVi. LetM :=i∈IGki(Vi) and fori∈I, let fi:M→(Vi)ki_{be a}

continuous mapping. Then, there exists ¯x=( ¯xi)i∈I∈M such that

∀i∈I, fi( ¯x)∈x¯iki. (3.1)

The proof ofTheorem 3.1is given in two steps. In the first step, we additionally assume that the mappings are smooth, and the second step gives the proof in the general case. 3.1.1. Proof ofTheorem 3.1when the fiare smooth. LetM :=i∈IGki(Vi) and define f :

M→i∈IVikiby f (x)=proj_(xki i )⊥ fi(x)  i∈I forx=  xi_i∈I∈M, (3.2) and the subsetsZ, Z1, andZ2ofM×i∈IVikiby

Z=  (x, y)∈M× i∈I Vki i | ∀i∈I, yi∈xiki⊥ , Z1=  (x, y)∈M× i∈I Vki i |y=f (x) , Z2=  (x, y)∈M× i∈I Vki i |y=0 . (3.3)

ProvingTheorem 3.1amounts to showing the existence of ¯x∈M such that f (¯x)=0 or, equivalently, such thatZ1∩Z2= ∅. For this, we will use the followingIntersection

Theorem 3.2, which is a direct consequence of mod 2 intersection theory (see, e.g., [10,

Intersection theorem 3.2. LetZ be a smooth boundaryless manifold of dimension 2m and letZ1,Z2 be two compact boundaryless submanifolds ofZ of dimension m. If ¯Z1 is a compact boundaryless m-submanifold of Z homotopic to Z1 and if the manifolds ¯Z1 and Z2intersect transversally in a unique point ¯z (which means that Tz¯Z¯1+Tz¯Z2=Tz¯Z), then Z1∩Z2= ∅.

The proof ofTheorem 3.1consists of checking that the above-defined setsZ, Z1, and Z2 (together with the set ¯Z1 defined below) satisfy the assumptions of Intersection

Theorem 3.2.

The setsZ, Z1, andZ2 satisfy the assumptions ofIntersection Theorem 3.2. We recall that for everyi∈I, Gki(V_{i) is a smooth, boundaryless, compact manifold of dimension}

ki(dimVi−ki) (seeLemma A.1in the appendix). ThusM :=i∈IGki(Vi) is a

bound-aryless, smooth, compact manifold of dimensionm=i∈Iki(dimVi−ki). ClearlyZ is

a fiber bundle whose base space isM and whose fiber at x=(xi)i∈I∈M is the vector

spacei∈I(xkii)⊥which has the dimension ofM. Hence, Z is a smooth manifold of

di-mension 2m.

The mapping f : M→i∈IViki is a smooth mapping from Parts (c), (d), and (e) of

Lemma A.1in the appendix. Consequently,Z1 is a smooth compact boundaryless

sub-manifold of Z of dimension m. Finally, Z2 is clearly a smooth boundaryless compact submanifold ofZ of dimension m.

The manifoldZ1 is homotopic to the manifold ¯Z1that we now define. For everyi∈I, let ¯xi∈Gki(V_{i) and let} {e_¯1

i,..., ¯ekii} be an orthonormal basis of ¯xi. For every i∈I, let gi:Gki(V_i)→V_iki _andg : M_→

i∈IVikibe the mappings defined as follows:

∀xi∈GkiVi, gixi=projxi⊥  ¯ e1 i,...,projxi⊥  ¯ eki i ∈x⊥i ki=xkii⊥, ∀x=xii∈I∈M, g(x)=gixii∈I. (3.4) We let ¯ Z1:=  (x, y)∈M× i∈I Vki i |y=g(x) . (3.5)

To show that the manifoldZ1 is homotopic to ¯Z1, we letH : [0,1]×Z1→Z be the continuous mapping defined byH(t,(x, f (x)))=(x,(1−t) f (x) + tg(x)). Then H(0,·) is the canonical inclusion fromZ1toZ, and H(1,·)(Z1)=Z¯1.

The manifolds ¯Z1 andZ2 intersect transversally in a unique point. First, notice that ¯

Z1∩Z2= {(x,0)∈M×i∈IViki|g(x)=0}is the singleton ( ¯x,0)=(( ¯xi)i∈I, 0). But that

¯

Z1andZ2intersect each other transversally inZ means that T( ¯x,0)Z¯1+T( ¯x,0)Z2=T( ¯x,0)Z.

Recalling that dimT( ¯x,0)Z¯1+ dimT( ¯x,0)Z2=dimT( ¯x,0)Z=2m, we only have to show that T( ¯x,0)Z¯1∩T( ¯x,0)Z2= {0}. Finally, noticing thatT( ¯x,0)Z¯1= {(u,Dg(¯x)(u))|u∈Tx¯M}and T( ¯x,0)Z2= {(u,0)|u∈Tx¯M}, we only have to prove that Dg(¯x) is injective, which is proved in the following lemma.

Lemma 3.3. Dg(¯x) is injective.

Proof. Recalling that for everyx=(xi)i∈I∈M, g(x)=(gi(xi))i∈I, the mappingDg(¯x) is injective if and only if for everyi∈I, Dgi( ¯xi) is injective.

So, leti∈I, let (ϕ,U) be a local chart of Gki(V_{i) at ¯}x_{i, and let}ψ : (¯x⊥

i )ki→Gki(Vi) be the inverse mapping ofϕ : U→( ¯xi⊥)ki. From the appendix, if {e¯11,..., ¯e

ki

i }is a given

orthonormal basis of ¯xi,ψ can be defined by

ψu1_,...,uki_=spane_¯1

i +u1,..., ¯eiki+uki for everyu1,...,uki∈x¯⊥i ki. (3.6)

Since the mappinggi◦ψ is the local representation giin the chart (ϕ,U), proving that

Dgi( ¯xi) is injective amounts to proving thatD(gi◦ψ)(0) is injective. This is a consequence

of the following claim.

Claim 3.4. For all (h1,...,hki)∈( ¯x⊥i )ki,D(gi◦ψ)(0)(h1,...,hki)= −(h1,...,hki).

Proof ofClaim 3.4. Letp : Vi×( ¯x⊥i )ki→Vibe defined by

p(y,u) :=proj_ψ(u)y. (3.7)

If we prove that for everyy∈Vi, the derivative of the mappingpy:u→p(y,u) is the linear mappingDpy(0) : ( ¯x⊥i )ki→Videfined by

Dpy(0)(h)= ki  k=1  y·e¯kihk, ∀h=h1,...,hki  ∈x¯⊥i ki, (3.8)

thenClaim 3.4 will be proved. Indeed, taking y=e¯_ik for every k=1,...,ki, we would

obtainDe¯k

ip(0)(h1,...,hki)=hk. Thus, sincegi◦ψ(u)=( ¯e

1

i,..., ¯ekii)−(pe¯1

i(u),..., pe¯kii (u)),

it would entailClaim 3.4.

Now, for everyu=(u1,...,uki)∈( ¯xi⊥)ki, there existsλ(y,u)=(λk(y,u))k=1,...,ki∈Rki

such that

p(y,u)=proj_ψ(u)y= ki

k=1

λk(y,u)e¯ki+uk, (3.9)

with (λk(y,u)) satisfying

 −y +ki k=1 λk(y,u)e¯ik+uk ·e¯ij+uj=0 for everyj=1,...,ki. (3.10)

This can be equivalently rewritten as follows: 

Iki+G(u)



λ(y,u)=y·e¯1i+u1,..., y·e¯kii+uki, (3.11)

where Iki is the ki×ki identity matrix and G(u) is the ki×ki matrix G(u)=

(uj_·uk_)j,k

=1,...,ki. Besides, foru in a neighborhood ᏺ of 0 small enough, the matrix (Iki+

that the mappingp(·,·) is smooth onV×ᏺ. Differentiating, with respect to u, the above equality atu=0, we obtain, for everyh=(h1,...,hki)∈( ¯x⊥i )ki,

DG(0)(h)λ(y,0) + Duλ(y,0)(h)=0. (3.12) But it is clear thatDG(0)=0. Consequently,Duλ(y,0)=0.

Finally, differentiating the equality p(y,u)=ki

k=1λk(y,u)(¯eki +uk) at (y,0), one ob-tains, for everyh=(hk)kki=1∈( ¯xi⊥)k,

Dup(y,0)(h)= ki  k=1 λk(y,0)hk= ki  k=1  y·e¯ikhk, (3.13)

which ends the proof ofClaim 3.4.

3.1.2. Proof ofTheorem 3.1in the general case. SinceM is a compact manifold and Vki

i is

a Euclidean space, for everyi∈I, each continuous mapping fi:M→Vikican be

approx-imated by a sequence of smooth mappings fin:M→Viki converging to fi, in the sense

that limn→∞fin−fi∞=0 (see, e.g., Hirsch [12]). Applying the first step to the smooth mappings fin, we deduce the existence of (xni)i∈I∈M such that

∀i∈I, fn i xni∈xinki (3.14) or, equivalently, proj_(xn⊥ i )ki fi  xn i=0. (3.15)

From the compactness ofM, without any loss of generality, one can suppose that the sequence (xn

i)i∈Iconverges to some element ( ¯xi)i∈I∈M. We have

_proj ( ¯xi⊥)ki fi  ¯ xi−proj( ¯xn⊥ i )ki f n i xni ≤proj_{( ¯xi}⊥₎ki fix¯i−proj( ¯xni⊥)ki fi  xn i+fin−fi_∞. (3.16)

Consequently, from the convergence of fin to fi and the continuity of the mapping

(u,v)→proj(u⊥)kiv (seeLemma A.1in the appendix), we obtain

proj_{( ¯xi}⊥₎kifix¯i=0 (3.17)

or, equivalently,

∀i∈I, fix¯i∈x¯⊥i ki

⊥

=x¯iki, (3.18) which ends the proof ofTheorem 3.1.

3.2. Proof ofTheorem 2.1whenJ= ∅. We now proveTheorem 2.1whenJ= ∅. The proof rests on the following claim.

Claim 3.5. For everyi∈I and every k∈ {1,...,ki}, there exists an u.s.c. correspondence ˆFik

fromM to Vi, with nonempty convex values, such that ∀x∈M, Fk

i(x)= ∅=⇒∀y∈Fˆik(x),∃λ∈R,λy∈Fik(x). (3.19)

Proof ofClaim 3.5. Leti∈I and k∈ {1,...,ki}. We distinguish two cases.

Assume first thatFikis l.s.c. LetUik= {x∈M|Fik(x)= ∅}. ThenUikis an open subset

ofM and Fik|Uk

i is a l.s.c. correspondence with nonempty convex values. By Michael [15],

there exists a continuous selectionfikofFik|Uk

i, that is, f

k

i :Uik→Viis a continuous

map-ping such thatfik(x)∈Fik(x) for every x∈Uik. LetBibe the closed unit ball ofVi, and we define the correspondence ˆFikfromM to Biby ˆFik(x)= {fik(x)/fik(x)}ifx∈Uik and

fk

i (x)=0 and ˆFik(x)=Biotherwise. We let the reader check that the correspondence ˆFik

satisfies the conclusion ofClaim 3.5.

We now consider the case whereFikis u.s.c. LetUik= {x∈M|Fik(x)= ∅}.ThenUik

is a closed subset ofM. By Cellina [4], one can extendFik|Ui as follows: there exists a

correspondence ˆFikfromM to Viwhich is u.s.c., with nonempty, convex, and compact

values, such that for everyx∈Uk

i,Fik(x)=Fˆik(x).

We now come back to the proof of Theorem 2.1when J= ∅. For every i∈I and k=1,...,ki, let ˆFik be the u.s.c. correspondence fromM to Vi with nonempty convex (compact) values defined inClaim 3.5. By Cellina [3], for every integern, there exists a continuous mapping f_ik,n:M→Visuch that

Gfik,n⊂GFˆik+B

 0,1_n



. (3.20)

Now, fori∈I, let fin:M→(Vi)ki_{be defined as follows:}

∀x∈M, fn

i (x)=fi1,n(x),..., fiki,n(x). (3.21)

ApplyingTheorem 3.1to the mappings fin, we deduce the existence of ( ¯xni)i∈I∈M

such that for everyi∈I, fin( ¯xn)∈( ¯xin)ki, hence

yik,n:=fik,nx¯n∈x¯in. (3.22)

Since the correspondence ˆFik is bounded (M is compact and ˆFik is u.s.c.), the sequence

(yk,ni ) is bounded. Thus, without any loss of generality, one can suppose that the sequence

(yk,ni ) converges to someyki ∈Viwhenn tends to +∞.

Besides, from the compactness ofM, without any loss of generality, one can suppose that ( ¯xni)i∈Iconverges to ¯x=( ¯xi)i∈I∈M when n tends to +∞.

Moreover, fromLemma A.1(d) in the appendix and fromyik,n∈x¯k,ni , at the limit we

have that

Since the graph of ˆFikis closed (it is u.s.c. with compact values) and fromG( fik,n)⊂

G( ˆFk

i) +B(0,1/n), one obtains

yk

i ∈Fˆik( ¯x). (3.24)

To end the proof, we assume thatFik( ¯x)= ∅. Since yik∈Fˆik( ¯x), byClaim 3.5, there

exists λ∈Rsuch thatλyki ∈Fik( ¯x). Hence λyik∈Fik( ¯x)∩x¯i= ∅ (since yik∈x¯i). This ends the proof ofTheorem 2.1.

3.3. Proof ofTheorem 2.1in the general case. We first prove the following lemma.

Lemma 3.6. LetC be a nonempty, convex, compact subset of a Euclidean space V. Then there exists a continuous mappingρ : G1_{(V×R)→C such that}

∀x∈G1_{(V×R), x∩}_{C× {1}}_{= ∅ =⇒x∩}_{C× {1}}₌_ρ(x),1_{. (3.25)} Proof. SinceC is compact, it is included in a closed ball B(0,k) of V. We let r : V →

B(0,k + 1) be defined by r(u)=α(u)u, where α :R+→R+is defined by α(t)=1 ift∈[0,k],

α(t)=k + 1−t if t∈[k,k + 1], α(t)=0 ift≥k + 1.

(3.26)

Letπ1:V×R→V and ρ : G1(V×R)→C be defined by π1(x,t)=x and ρ(x)=    proj_C◦r◦π1x∩V× {1} ifx∩V× {1}= ∅, projC(0) ifx∩V× {1}= ∅, (3.27) where proj_C:V →C denotes the projection from V to C. Then, one easily sees that ρ

satisfies the conclusion ofLemma 3.6.

Proof ofTheorem 2.1. UsingLemma 3.6, we first modify the correspondencesFjfor ev-ery j∈J and replace each nonempty compact convex set Cj⊂Vj by the Grassmannian

manifoldG1_(V

j×R). For everyj∈J, let ρj:G1(Vj×R)→Cjbe the mapping associated

toCj⊂VjbyLemma 3.6. Let ρ : ˜M := i∈I GkiV i× j∈J G1_V j×R−→M := i∈I GkiV i× j∈J C j (3.28) be defined by ρ(x)=xii∈I,ρjxjj∈J  , forx=xii∈I,xjj∈J  . (3.29)

Fori∈I and k=1,...,ki, let ˜Fikbe the correspondence from ˜M to Videfined by ˜

Fk

For j∈J, let ˜Fjbe the correspondence from ˜M to Vj×Rdefined by ˜

Fj(x)=Fjρ(x)× {1}. (3.31)

Now, applying the result proved inSection 3.2 (i.e.,Theorem 2.1 when J= ∅) to the correspondences ˜Fikand ˜Fj, there existsx=((xi)i∈I, (xj)j∈J)∈M such that˜

(i) either ˜Fik(x)∩xi= ∅or ˜Fik(x)= ∅for everyi∈I and i=1,...,ki, (ii) either ˜Fj(x)∩xj= ∅or ˜Fj(x)= ∅for everyj∈J.

Let ¯x=ρ(x)∈M; we end the proof by showing that it satisfies the conclusion ofTheorem 2.1. From the above, it is clearly the case fori∈I and k=1,...,ki, that is, we have that

(i) eitherFik(x)∩xi= ∅orFik(x)= ∅for everyi∈I and i=1,...,ki.

Now, let j∈J. We first notice that ˜Fj(x)= ∅if and only ifFj(x)= ∅. Assume now that ˜

Fj(x)∩xj= ∅and recall thatxj∩F˜j(x)=xj∩(Fj( ¯x)× {1}) andFj( ¯x)⊂Cj. Conse-quently,xj∩(Cj× {1})= ∅and fromLemma 3.6we get

∅ =xj∩Fj( ¯x)× {1}⊂xj∩Cj× {1}=ρjxj, 1. (3.32)

Hence, the equality holds and ¯xj=ρj(xj)∈Fj( ¯x). This ends the proof ofTheorem 2.1.

Appendix

The Grassmannian manifoldGk(V)

LetV be a Euclidean space and let k be an integer such that 0≤k≤dimV. In this section, we recall the properties ofGk(V) which are used in this paper.

First, we recall thatGk(V) is a smooth boundaryless manifold of dimension k(dimV− k) (see, e.g., Hirsch [12] andLemma A.1). The local charts can be defined as follows. Let ¯E∈Gk_{(V) and let{e¯}

1,..., ¯ek}be some given orthonormal basis of ¯E; we define the

mappingψ_E¯: ( ¯E⊥)k→Gk(V) by

ψ_E¯(u)=spane¯1+u1,..., ¯ek+uk, foru=u1,...,uk∈E¯⊥k. (A.1)

Then it is easy to check that the mapping ψ_E¯ is injective (see Claim A.2); so ψ_E¯ is a bijection from ( ¯E⊥)k ontoU_E¯=ψ_E¯(( ¯E⊥)k). We can now consider the inverse mapping ϕ_E¯:U_E¯→( ¯E⊥)kdefined byϕ_E¯(E)=ψ_E−_¯1(E), which is clearly a bijection.

Lemma A.1. (a) Gk(V) is a smooth boundaryless (i.e., C∞) manifold of dimension k(dimV−k) without boundary and (UE¯,ϕE¯)E¯∈Gk_(V)defines aC∞atlas ofGk(V).

(b) The setGk(V) is compact.

(c) The mappingE→E⊥fromGk(V) to G(V) (=dimV−k) is a smooth diffeomor-phism.

(d) The mapping p : V×Gk_{(V)→V defined by p(x,E)=proj}

E(x) is smooth. Hence,

the set{(x,E)∈V×Gk(V)|x∈E}is a closed subset ofV×Gk(V). (e) The mappingx→xp_fromGk_{(V) to (G}k_(V))p_{is smooth.}

Claim A.2. Let ¯E∈Gk(V) and let{e¯1,..., ¯ek}be an orthonormal basis of ¯E. (a) The mappingψ_E¯is injective.

(b) For everyu∈( ¯E⊥)k,ψ(0)∩ψ(u)⊥_=ψ(0)⊥_{∩ψ(u)= {0}.}

Proof ofClaim A.2. Part (a). Letu=(u1,...,uk) andv=(v1,...,vk) in ( ¯E⊥)ksuch that

ψ_E¯(u)=spane¯1+u1,..., ¯ek+uk=ψ_E¯(v)=spane¯1+v1,..., ¯ek+vk. (A.2)

Then, there exist some real numbersλij(i=1,...,k, j=1,...,k) such that

¯ ei+ui=

k

j=1

λije¯j+vj, for everyi=1,...,k. (A.3)

Taking, for each inequality, the scalar product with ¯el, wherel=1,...,k, we obtain λil=0 ifi=l and λll=1. Henceul=vlfor everyl=1,...,k, and finally u=v.

Part (b). Letx∈_E¯_∩ψ(u)⊥_{. Then there exists some real numbersλi(i=1,...,k) such}

thatx=ki=1λie¯i. Taking the scalar product with ¯ej+uj(j=1,...,k), we obtain λj=0 for every j=1,...,k, which proves ψ(0)∩ψ(u)⊥_{= {0}. Similarly, letx∈E}¯⊥_{∩ψ(u). Then}

there exists some real numbersλi(i=1,...,k) such that x=ki=1λi( ¯ei+ui). Taking the scalar product with ¯ej (j=1,...,k), we obtain λj=0 for everyj=1,...,k, which proves

ψ(0)∩ψ(u)⊥_{= {0}and ends the proof of the claim.}

Proof ofLemma A.1. Part (a). We prove that (UE,ϕE)E∈Gk_(V)is a smooth (i.e.,C∞) atlas

ofGk(V) and it is then clear that dimM=dim(E⊥)k=k(dimV−k). Let (UE,ϕE) and (UF,ϕF) be two local charts atE and F, respectively, such that UE∩UF= ∅. We will prove

thatϕF◦ϕ−E1is smooth (i.e.,C∞). We let{e1,...,ek}and{f1,..., fk}be two orthonormal

bases ofE and F, respectively. Let (v1,...,vk)=ϕF◦ϕE−1(u1,...,uk) andu=(u1,...,uk); then there exist real numbersλij(u) (i, j=1,...,k) such that

fi+vi= k

j=1

λij(u)ei+ui (i=1,...,k). (A.4)

The proof will be complete by showing that the real-valued functionsλij(u) are

differen-tiable with respect tou. Taking the scalar product with fl(l=1,...,k), we obtain

fi·fl= k

j=1

λij(u)ej+uj·fl (l=1,...,k). (A.5)

Thus, for everyi=1,...,k, the vector λi(u)=(λij(u))kj=1is the solution of a linear sys-tem whose matrix G(u)=((ej+uj)·fl)j,l=1,...,k is now shown to be invertible (which

clearly implies the differentiability of λi(u)). Indeed, if G(u)λ=0 for someλ∈Rk, then

k

j=1λj(ej+uj)· fl=0 (for l=1,...,k), thus

k

j=1λj(ej+uj)∈F⊥. Besides, since

k

j=1λj(ej+uj)∈ψE(u1,...,uj)=ψF(v1,...,vj), we finally obtain kj=1λj(ej+uj)∈ F⊥_∩ψF(v

1,...,vk)= {0}(fromClaim A.2). Now, since (ej+uj)j=1,...,kis a basis, we ob-tainλj=0 for every j=1,...,k.

Part (b). Let (Eν_{) be a sequence in G}k_{(V) and for every ν, let{e}ν

1,...,eνk} be an

or-thonormal basis of Eν. Without any loss of generality, we can assume that for every i=1,...,k, the sequence (eνi) converges to some elementeiin V. Clearly,{e1,...,ek} is

an orthonormal family inV, and we let E=span{e1,...,ek}. We will now prove that the sequence (Eν_{) converges toE. Indeed, for ν large enough, there exists u}ν_=(uν

1,...,uνk)∈

(E⊥₎k_{such thatE}ν_=ψE(uν

1,...,uνk). It can be written asei+uνi =kj=1λνijeνj (i=1,...,k). Multiplying these equalities byel(l=1,...,k), we obtainkj=1λνijeνj·el=0 ifi=l and k

j=1λνijeνj·ei=1. This can be written (for everyi=1,...,k) as (eνj·el)j,l=1,...,k(λνij)kj=1= (ei·el)l=1,...,k. Ifν is large enough, then (eνj·el)j,l=1,...,kis invertible and converges to Id,

which proves that the sequence (λν

ij)kj=1 converges to (ei·el)l=1,...,k for everyi=1,...,k, that is, (λν

ii) converges to 1 and (λνij) converges to 0 fori=j. We finally obtain that (uνi)

converges to 0, which proves thatEνconverges toE.

Part (c). Let ¯E∈Gk(V) and let (¯e1,..., ¯ek) and ( ¯f1,..., ¯f) be orthonormal bases of ¯

E and ¯E⊥_{, respectively. Let (u}

1,...,uk)∈( ¯E⊥)k and let E=ψ(u). Then it is easy to see thatE⊥=span{f1+v1,..., f+v}, wherevi= −

k

j=1(fi·uj) ¯ej. So eachviis a smooth mapping with respect to the ui and, conversely, from (E⊥)⊥=E, each ui is a smooth

mapping with respect to thevi. This ends the proof of part (c).

Part (d). The differentiability of the mapping p is left to the reader. Then notice that

{(x,E)∈V ×Gk_{(V)|x∈E} = {(x,E)∈V×G}k_(V)|x=proj

E(x)}, which is clearly

closed since the mappingp : (x,E)→proj_E(x) is continuous.

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Philippe Bich: Centre de Recherche en Math´ematiques de la D´ecision, Universit´e Paris-Dauphine, Place du Mar´echal de Lattre de Tassigny, 75775 Paris, France

E-mail address:[email protected]

Bernard Cornet: Centre de Recherche en Math´ematiques, Statistique et Economie Math´ematique, Universit´e Paris 1, 106-112 boulevard de l’H ˆopital, 75647 Paris, France