FIXED-POINT-LIKE THEOREMS ON SUBSPACES
PHILIPPE BICH AND BERNARD CORNETReceived 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. Letkbe an in-teger and letV be a Euclidean space such that 0≤k≤dimV, then thek-Grassmannian manifold ofV, denotedGk(V), is the set of all thek-dimensional subspaces ofV. 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
and the appendix recalls the main properties of the Grassmannian manifold, used in this paper.
2. Statement of the results
2.1. Preliminaries. A correspondenceΦfrom a setXto a setYis a map fromXto the set of all the subsets ofY, and the graph ofΦ, 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 ofΦifϕ(x)∈Φ(x) for allx∈X. IfAis a subset ofX, we letΦ(A)=x∈AΦ(x), and the restriction ofΦtoA, denotedΦ|A, is the correspondence fromAtoY defined byΦ|A(x)=Φ(x) ifx∈A. IfX andY are topological spaces, the correspondenceΦis 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 inXand, 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∈Rn| x−y ≤r}. IfE is a vector subspace ofRn, we denote byE⊥= {u∈Rn| ∀x∈E, x·u=0}the orthogonal space toE. Ifu1,...,uk belong toE, a vector space, we denote by span{u1,...,uk}the vector subspace ofEspanned byu1,...,uk.
LetVbe a Euclidean space and letkbe an integer such that 0≤k≤dimV; we denote byGk(V) the set consisting of all the linear subspaces ofV of dimensionk, called the (k-)Grassmannian manifold ofV. 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.
Theorem2.1. LetI,Jbe two finite disjoint sets. For everyi∈I, letkibe an integer and let Vi be a Euclidean space such that0≤ki≤dimVi. For every j∈J, letCj be a nonempty, convex, compact subset of a Euclidean spaceVj, and letM=i∈IGki(Vi)×j∈JCj.
Fori∈Iandk=1,...,ki, letFikbe a correspondence fromM toViwith convex values, for j∈J, letFjbe a correspondence fromMtoCjwith convex values, and suppose that, for everyi∈Iandk=1,...,ki(resp., j∈J), the correspondenceFik(resp.,Fj) is either l.s.c or u.s.c.
Then, there existsx¯=(( ¯xi)i∈I, ( ¯xj)j∈J)∈Msuch that
(i)eitherFik( ¯x)∩x¯i= ∅orFik( ¯x)= ∅for everyi∈Iandk=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].
Corollary2.2. LetV1be a Euclidean space, letk1be an integer such that0≤k1≤dimV1, and for everyk=1,...,k1, letfk:Gk1(V1)→V1be a continuous mapping. Then, there exists
¯
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.
Corollary2.3. LetJbe a finite set, forj∈J, letCjbe 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 existsx¯=( ¯xj)j∈J∈ Msuch that for everyj∈J, eitherx¯j∈Fj( ¯x)orFj( ¯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 elementDofG1(R2) can be written as Dt= {λ(cost, sint)|λ∈R}, for somet∈[0,π[. We define the correspondenceF from M toR2 byF(D0)=R× {1}andF(D
t)=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 thatF 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.
Corollary2.5. Fork=1,...,n, letFkbe 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∈Snsuch 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∈Fk(−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∈Snsuch that 0∈Fˆk( ¯x) for everyk=1,...,n.
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∈Gn(Rn+1) such that ¯E∩Hk( ¯E)= ∅for everyk=1,...,n.
Now, if ¯xis an arbitrary point of ¯E⊥∩Sn, then we have ¯E∩Fˆk( ¯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.
Theorem3.1. LetIbe a finite set and fori∈I, letkibe an integer and letVibe a Euclidean space such that0≤ki≤dimVi. LetM:=i∈IGki(Vi)and fori∈I, let fi:M→(Vi)kibe a
continuous mapping. Then, there existsx¯=( ¯xi)i∈I∈Msuch 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=
xii∈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)
Intersection theorem3.2. LetZ be a smooth boundaryless manifold of dimension2m and letZ1,Z2 be two compact boundaryless submanifolds ofZof dimensionm. IfZ1¯ is a compact boundaryless m-submanifold ofZ homotopic toZ1 and if the manifolds Z1¯ and Z2intersect transversally in a unique pointz¯(which means thatT¯zZ1¯ +T¯zZ2=T¯zZ), 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(Vi) 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 atx=(xi)i∈I∈M is the vector spacei∈I(xkii)⊥which has the dimension ofM. Hence,Zis 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 ofZof dimensionm.
The manifoldZ1 is homotopic to the manifoldZ1¯ that we now define. For everyi∈I, let ¯xi∈Gki(Vi) and let {e¯1
i,..., ¯ekii} be an orthonormal basis of ¯xi. For every i∈I, let gi:Gki(Vi)→Viki 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)). ThenH(0,·) is the canonical inclusion fromZ1toZ, andH(1,·)(Z1)=Z1¯ .
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 inZmeans thatT( ¯x,0)Z1¯ +T( ¯x,0)Z2=T( ¯x,0)Z. Recalling that dimT( ¯x,0)Z1¯ + dimT( ¯x,0)Z2=dimT( ¯x,0)Z=2m, we only have to show that T( ¯x,0)Z1¯ ∩T( ¯x,0)Z2= {0}. Finally, noticing thatT( ¯x,0)Z1¯ = {(u,Dg( ¯x)(u))|u∈T¯xM}and T( ¯x,0)Z2= {(u, 0)|u∈T¯xM}, we only have to prove that Dg( ¯x) is injective, which is proved in the following lemma.
Lemma3.3. Dg( ¯x)is injective.
So, leti∈I, let (ϕ,U) be a local chart ofGki(Vi) at ¯xi, and letψ: ( ¯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,ψcan 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 representationgiin 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.
Claim3.4. 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) :=projψ(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 obtainD¯ek
ip(0)(h1,...,hki)=hk. Thus, sincegi◦ψ(u)=( ¯e
1
i,..., ¯ekii)−(p¯e1
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)=projψ(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¯iki+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, foruin a neighborhoodᏺof 0 small enough, the matrix (Iki+
that the mappingp(·,·) is smooth onV×ᏺ. Differentiating, with respect tou, 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)=kk=i 1λk(y,u)( ¯eki +uk) at (y, 0), one ob-tains, for everyh=(hk)k=ki 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. SinceMis a compact manifold andVki
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∈Msuch that
∀i∈I, finxin∈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 (xni)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)
3.2. Proof ofTheorem 2.1whenJ= ∅. We now proveTheorem 2.1whenJ= ∅. The proof rests on the following claim.
Claim3.5. For everyi∈Iand everyk∈ {1,...,ki}, there exists an u.s.c. correspondenceFˆik fromMtoVi, with nonempty convex values, such that
∀x∈M, Fik(x)= ∅=⇒∀y∈Fˆik(x), ∃λ∈R,λy∈Fik(x). (3.19)
Proof ofClaim 3.5. Leti∈Iandk∈ {1,...,ki}. We distinguish two cases.
Assume first thatFikis l.s.c. LetUik= {x∈M|Fik(x)= ∅}. ThenUikis an open subset ofMandFik|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 everyx∈Uik. LetBibe the closed unit ball ofVi, and we define the correspondence ˆFikfromM toBiby 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 toViwhich 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 toVi with nonempty convex (compact) values defined inClaim 3.5. By Cellina [3], for every integern, there exists a continuous mapping fik,n:M→Visuch that
Gfk,n
i ⊂GFˆik+B
0,1n
. (3.20)
Now, fori∈I, let fin:M→(Vi)kibe defined as follows:
∀x∈M, fin(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
yk,n
i :=fik,nx¯n∈x¯in. (3.22)
Since the correspondence ˆFik is bounded (M is compact and ˆFik is u.s.c.), the sequence (yki,n) is bounded. Thus, without any loss of generality, one can suppose that the sequence (yki,n) converges to someyik∈Viwhenntends to +∞.
Besides, from the compactness ofM, without any loss of generality, one can suppose that ( ¯xni)i∈Iconverges to ¯x=( ¯xi)i∈I∈Mwhenntends to +∞.
Moreover, fromLemma A.1(d) in the appendix and fromyik,n∈x¯ki,n, 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( ˆFik) +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. Lemma3.6. LetCbe a nonempty, convex, compact subset of a Euclidean spaceV. Then there exists a continuous mappingρ:G1(V×R)→Csuch 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) ofV. We let r:V → B(0,k+ 1) be defined byr(u)=α(u)u, whereα:R+→R+is defined by
α(t)=1 ift∈[0,k],
α(t)=k+ 1−t ift∈[k,k+ 1], α(t)=0 ift≥k+ 1.
(3.26)
Letπ1:V×R→Vandρ:G1(V×R)→Cbe defined byπ1(x,t)=xand
ρ(x)=
projC◦r◦π1x∩V× {1} ifx∩V× {1}= ∅,
projC(0) ifx∩V× {1}= ∅,
(3.27)
where projC:V →Cdenotes the projection from V toC. Then, one easily sees thatρ
satisfies the conclusion ofLemma 3.6.
Proof ofTheorem 2.1. UsingLemma 3.6, we first modify the correspondencesFjfor ev-ery j∈Jand replace each nonempty compact convex setCj⊂Vj by the Grassmannian manifoldG1(V
j×R). For everyj∈J, letρj:G1(Vj×R)→Cjbe the mapping associated toCj⊂VjbyLemma 3.6. Let
ρ: ˜M:= i∈I
GkiV
i× j∈J
G1V
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∈Iandk=1,...,ki, let ˜Fikbe the correspondence from ˜MtoVidefined by
˜ Fk
For j∈J, let ˜Fjbe the correspondence from ˜MtoVj×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∈Iandi=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∈Iandk=1,...,ki, that is, we have that
(i) eitherFik(x)∩xi= ∅orFik(x)= ∅for everyi∈Iandi=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)
LetVbe a Euclidean space and letkbe 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 dimensionk(dimV− k) (see, e.g., Hirsch [12] andLemma A.1). The local charts can be defined as follows. Let ¯E∈Gk(V) and let{e1¯ ,..., ¯ek}be some given orthonormal basis of ¯E; we define the mappingψE¯: ( ¯E⊥)k→Gk(V) by
ψE¯(u)=spane1¯ +u1,..., ¯ek+uk, foru=u1,...,uk∈E¯⊥k. (A.1)
Then it is easy to check that the mapping ψE¯ is injective (see Claim A.2); so ψE¯ is a bijection from ( ¯E⊥)k ontoUE¯=ψE¯(( ¯E⊥)k). We can now consider the inverse mapping ϕE¯:UE¯→( ¯E⊥)kdefined byϕE(¯ E)=ψE−¯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)toG(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).
ClaimA.2. LetE¯∈Gk(V)and let{e1¯,..., ¯ek}be an orthonormal basis ofE¯. (a)The mappingψE¯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
ψE¯(u)=spane1¯ +u1,..., ¯ek+uk=ψE¯(v)=spane1¯ +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 obtainλil=0 ifi=landλll=1. Henceul=vlfor everyl=1,...,k, and finallyu=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 obtainλj=0 for every j=1,...,k, which provesψ(0)∩ψ(u)⊥= {0}. Similarly, letx∈E¯⊥∩ψ(u). Then there exists some real numbersλi(i=1,...,k) such thatx=ki=1λi( ¯ei+ui). Taking the scalar product with ¯ej (j=1,...,k), we obtainλj=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 atEandF, respectively, such thatUE∩UF= ∅. We will prove thatϕF◦ϕ−E1is smooth (i.e.,C∞). We let{e1,...,ek}and{f1,...,fk}be two orthonormal bases ofEandF, 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 diff eren-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 vectorλi(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 ofλi(u)). Indeed, ifG(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 k
j=1λj(ej+uj)∈ F⊥∩ψF(v1,...,vk)= {0}(fromClaim A.2). Now, since (ej+uj)j=
Part (b). Let (Eν) be a sequence in Gk(V) and for everyν, let{e1ν,...,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 letE=span{e1,...,ek}. We will now prove that the sequence (Eν) converges toE. Indeed, forνlarge enough, there existsuν=(uν1,...,uνk)∈ (E⊥)ksuch thatEν=ψE(uν1,...,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=land 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. Ifνis large enough, then (eνj·el)j,l=1,...,kis invertible and converges to Id, which proves that the sequence (λνij)kj=1 converges to (ei·el)l=1,...,kfor everyi=1,...,k, that is, (λνii) 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 ¯
Eand ¯E⊥, respectively. Let (u1,...,uk)∈( ¯E⊥)k and let E=ψ(u). Then it is easy to see thatE⊥=span{f1+v1,...,f+v}, wherevi= −kj=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 mappingpis left to the reader. Then notice that {(x,E)∈V ×Gk(V)|x∈E} = {(x,E)∈V×Gk(V)|x=proj
E(x)}, which is clearly closed since the mappingp: (x,E)→projE(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
