CM 3 : One dimensional Convex Integration

CM 3 : One dimensional Convex Integration

Vincent Borrelli November 20, 2012

Convex Integration Theory is a powerful tool for solving differential re- lations. It was introduced by M. Gromov in his thesis dissertation in 1969, then published in an article [2] in 1973 and eventually generalized in a book [3] in 1986. Nevertheless, reading Gromov is often a challenge since im- portant details are not provided explicitely. Fortunately, there is a good reference that leaves no details in the shadow : the Spring’s book [5]. My un- derstanding of Convex Integration Theory primarily comes from this book.

I owe it much in this presentation.

1 Two introductory examples

1.1 A first example

Let us consider the following elementary problem.

Problem 1.– Let

f₀ : [0,1] −→ R³ t 7−→ (0,0, t)

be the linear application mapping the segment [0,1] vertically in R³. The problem is to find f : [0,1]−→^C1 R³ such that:

i) ∀t∈[0,1], |cos(f⁰(t), e₃)|<

ii) kf−f0k_C0 < δ

where >0 andδ >0 are given.

Solution.– At a first glance, the problem seems hopeless since conditioni says that the slope is small and then the image has to move far away from the segment before reaching the desired height. After a few seconds of extra thinking, the solution occurs. It is good enough to move along an helix

spiralling around the vertical axis:

f : [0,1] −→ R³

t 7−→







δcos 2πN t δsin 2πN t t

whereN ∈N^∗ is the number of spirals. We have f⁰

kf⁰k, e₃

= 1

√

1 + 4π²N²δ².

Therefore, ifN is large enough,f fulfills conditionsi andii.

The image of f₀ is the green vertical segment, the solution f is the red helix.

Rephrasing.– The above problem was pretty easy, it will become very informative with a rephrasing of the two conditions. Condition (i) means that the image off⁰ lies inside the cone:

R={v∈R³\ {O} |

h v

|v|, e₃i

< } ∪ {O}.

By extension, that coneRis called the differential relationof our problem.

f ’

f ’

The coneRis pictured in blue, the image off⁰ is the red circle and the constant image off₀⁰ the green point outside the cone.

The C⁰-closeness required in the second condition, is a consequence of a geometric property of the derivative of f. Indeed, the image of f⁰ in that cone is a circle whose center is the constant image of f₀⁰. Therefore, the average off⁰ for each spiral of f isf₀⁰(t):

1 Long(Ik)

Z

Ik

f⁰(u)du=f₀⁰(t)

where I_k = [_N^k,^k+1_N ] the preimage of one spiral by f. Therefore, when integrating, the two resulting maps are closed together.

1.2 An more general example

Problem.– Let R ⊂ R³ be a path-connected subset (=our differential relation) andf₀ : [0,1]−→^C1 R³ be a map such

∀t∈[0,1], f₀⁰(t)∈ IntConv(R)

whereIntConv(R) denotes the interior of the convex hull ofR. The prob- lem is to findf : [0,1]−→^C1 R³ such that :

i) ∀t∈[0,1], f⁰(t)∈ R ii) kf−f0k_C0 < δ withδ >0 given.

Solution.– From the hypothesis, the image off₀⁰ lies in the convex hull of R.The idea is to build f⁰ with an image lying inside R and such that, on average, it looks like the derivative off₀. One way to do that is to choose a thef⁰-image to resemble to a kind of spring. In the spring, each arc as the same effect, on average, as a small piece of the image of the initial mapf₀⁰. So, when integrating, the resulting map will be close to the initial map. As before, we will improve the closeness off to f0 by incresing the number of spirals.

f ’

0

R

f ’

The green bended spaghetti¹ pictures the image off₀⁰, the half of a spring in rep/pink is the chosen image for f⁰.

To formally construct a solution f of the problem, it is enough to choose a continuous family of loops ofR:

h: [0,1] −→ C⁰(R/Z,R)

u 7−→ hu

such that

∀u∈ [0,1], Z

[0,1]

h_u(s)ds=f₀⁰(u) i.ethe average of the loop hu isf₀⁰(u).

h u

f ’ (u)0

s

u

The image of the loop hu.In that picture, this image is an arc. This loop is a round-trip starting at one of the endpoint of the arc and arriving at the same

endpoint.

Then, the mapf⁰ is extracted from that familly of loops by a simple diagonal process

∀t∈ [0,1], f⁰(t) :=ht({N t}) whereN ∈N^∗ and {N u}is the fractional part ofN t.

1Spaghetto ?

({Nu}) f’(u):=h u

s

u

The image of f⁰.

Eventually, it remains to integrate to obtain a solution to our problem:

f(t) :=f₀(0) + Z t

0

h_u({N u})du.

We say that f is obtained fromf₀ by aconvex integration process.

Example.– Let f0 : [0,1] −→ [0,1], t 7−→ (0,0,1), as in problem 1. We define a constant family of loops by putting

h: [0,1]×[0,1] −→ R

(t, s) 7−→ (−Rsin 2πs, Rcos 2πs,1) where

R={v∈R³\ {O} |

h v

|v|, e₃i

< } ∪ {O}

and R∈R^∗+ such that ^{√ 1}

R²+1 < . We obviously have Z 1

0

h(t, s)ds=f₀⁰(t) =e1.

Whence, the map f obtained from f₀ by a convex integration convex has the following expression

f(t) = f₀(0) + Z t

0

h_u({N u})du

= 0 + Z t

0

(−Rsin 2πN s, Rcos 2πN s,1)du

=

R

2πN cos 2πN s, R

2πN sin 2πN s, t

This is the solution f to the first introductory problem (δ= _2πN^R ).

2 Finding the loops

In the above problem, we were wilfully blind to the question of the existence of the family of loops (hu)u∈[0,1] needed to build the solution. We now deal with that issue.

Notation.– LetA ⊂Rⁿ and a∈A.We denote by IntConv(A, a) the in- terior of the convex hull of the connected component ofAto whichabelongs.

Definition.– A (continuous) loop g : [0,1] → Rⁿ, g(0) = g(1), strictly surrounds z∈Rⁿ if

IntConv(g([0,1]))⊃ {z}.

Fundamental Lemma.– Let R ⊂ Rⁿ be an open set, σ ∈ R and z ∈ IntConv(R, σ) There exists a loop h : [0,1] ^C

0

−→ R with base point σ that strictly surroundsz and such that:

z= Z 1

0

h(s)ds.

Proof.– Since z ∈IntConv(R, σ),there exists a n-simplex ∆ whose ver- ticesy0, ..., ynbelong toRand such thatzlies in the interior of ∆.Therefore, there also exist

(α₀, ..., α_n)∈]0,1[ⁿ⁺¹ such that

n

X

k=0

α_k = 1 andz=

n

X

k=0

α_ky_k. Every loopg: [0,1]→ Rwith base pointσ and passing throughy₀, ..., y_n satisfiesIntConv(g([0,1])⊃ {z} i. e.

g surroundsz.

In general

z6=

Z 1 0

g(s)ds.

Let s₀, ..., s_n be the times for whichg(s_k) =y_k and letf_k : [0,1]→R^∗+ be such that :

i)f_k < η₁ sur [0,1]\[s_k−η₂, s_k+η₂], ii)

Z 1 0

f_k= 1,

withη1, η2 two small positive numbers. We set:

z_k:=

Z 1 0

g(s)f_k(s)ds.

The number >0 being given, we can chooseη1, η2 such that:

∀k∈ {0, ..., n}, kz_k−g(s_k)k ≤. SinceR in open andz∈Int∆, for small enough we have

z∈IntConv(z₀, ..., z_n).

Therefore, there exist (p0, ..., pn)∈]0,1[ⁿ⁺¹ such that

n

X

k=0

p_k = 1 and:

z =

n

X

k=0

p_kz_k =

n

X

k=0

p_k Z ₁

0

g(s)f_k(s)ds

= Z 1

0

g(s)

n

X

k=0

p_kf_k(s)ds =

Z 1 0

g(s)ϕ⁰(s)ds where we have set

ϕ⁰(s) :=

n

X

k=0

p_kf_k(s) and

ϕ: [0,1] −→ [0,1]

s 7−→

Z s 0

ϕ(u)du.

We have ϕ⁰(s) > 0, ϕ(0) = 0, ϕ(1) = 1. Thus ϕ is a strictly increasing diffeomorphism of [0,1].Let us employ the change of coordinatess=ϕ⁻¹(t), that is t=ϕ(s),we have

dt=ϕ⁰(s)ds

therefore:

z= Z 1

0

g(s)ϕ⁰(s)ds= Z 1

0

g◦ϕ⁻¹(t)dt.

Thush=g◦ϕ⁻¹ is our desired loop.

Remark.– A priori h ∈ Ωσ(R), but it is obvious that we can choose h among ”round-trips”i.e the space:

Ω^AR_σ (R) ={h∈Ω_σ(R)| ∀s∈[0,1] h(s) =h(1−s)}.

The point is that the above space is contractible. For every u ∈ [0,1] we then denote byh_u: [0,1]−→ Rthe map defined by

h_u(s) =

h(s) if s∈[0,^u₂]∪[1−^u₂] h(u) if s∈[^u₂,1−^u₂].

This homotopy induces a deformation retract of Ω^AR_σ (R) to the constant map

σe: [0,1] −→ R s 7−→ σ.

Parametric version of the Fundamental Lemma. –LetP be a compact manifold, E =P ×Rⁿ −→^π P be a trivial bundle, and R ⊂E be a set such that

∀p∈P, R_p :=π⁻¹(p)∩ R is an open set of Rⁿ Let σ∈Γ(R) andz∈Γ(E) such that:

∀p∈P, z(p)∈IntConv(R_p, σ(p)).

Then, there exists h:P×[0,1] ^C

∞

−→ R such that:

h(.,0) =h(.,1) =σ∈Γ^∞(R), ∀p∈P, h(p, .)∈Ω^AR_σ(p)(R_p)

and

∀p∈P, z(p) = Z 1

0

h(p, s)ds.

Proof.– The proof is rather long and technical. The main problem is the following: the result of the previous lemma rests on the existence of points y0, ..., ynof Rsuch thatz∈ IntConv({y₀, ..., yn}).If we want to mimic the previous proof while adding, we need to be able to follow continuously the points over P,that is, we need to show the existence of (n+ 1) continuous maps y0, ..., yn:P −→Rⁿ such that

∀p∈P, z(p)∈ IntConv({y₀(p), ..., yn(p)}).

Locally, it is easy to obtain maps hU : U ×[0,1] ^C

∞

−→ R over open sets U, the true problem is to glue them together. In order to do that, we take advantage of the contractibility of the round-trip loops. The following sequence of pictures should be enlightning.

A homotopy among loops surroundingz and joining h_U (red) to h_V (blue).

We then obtain a globally defined continuous maph:P×[0,1] ^C

∞

−→ Rsuch that

∀p∈P, z(p)∈ IntConv(h(p,[0,1])) and

h(.,0) =h(.,1) =σ∈Γ^∞(R), ∀p∈P, h(p, .)∈Ω^AR_σ(p)(R_p).

It eventually remains to reparametrize the maph so that

∀p∈P, z(p) = Z 1

0

h(p, s)ds.

For more details, see [5] p. 29-31.

C^∞ parametric version of the Fundamental Lemma. – Let P be a compact manifold, E =P×Rⁿ−→^π P a trivial bundle and R ⊂E be a set such that

∀p∈P, R_p :=π⁻¹(p)∩ R is an open set of Rⁿ Let σ∈Γ^∞(R) and z∈Γ^∞(E) such that

∀p∈P, z(p)∈IntConv(R_p, σ(p)).

Then there exists h:P×[0,1] ^C

∞

−→ R such that

h(.,0) =h(.,1) =σ∈Γ(R), ∀p∈P, h(p, .)∈Ω^AR_σ(p)(R_p) and

∀p∈P, z(p) = Z ₁

0

h(p, s)ds.

Proof.– Let (ρ : [0,1] −→ R)_>0 be a sequence of mollifiers. For every p∈P we define aC^∞ map by the formula

h(p, .) : [0,1] −→ Rⁿ

t 7−→ (h(p, .)∗ρ)(t).

We set

z(p) :=

Z 1 0

h(p, t)dt and we defineH :P×R−→Rⁿ by

H(p, t) :=h(p, t) +z(p)−z(p).

We have

Z ₁

0

H(p, t)dt=z(p).

If is small enough, the image of the map t 7−→ H(p, t) lies inside R_p. Thanks to the compactness ofP the choice of the can be made indepen-

dently ofp∈P.

3 C

-density

LetR ⊂Rⁿ be a arc-connected subset, f₀∈C^∞(I,Rⁿ) be a map such that f₀⁰(I)⊂ IntConv(R).From theC^∞parametric version of the Fundamental Lemma there exists aC^∞-map h:I×E/Z−→ Rsuch that

∀t∈I, f₀⁰(t) = Z 1

0

h(t, u)du.

We set

∀t∈I, F(t) :=f₀(0) + Z t

0

h(s, N s)ds withN ∈N^∗.

Definition.– We say thatF ∈C^∞(I,Rⁿ) is obtained fromf0 by anconvex integration process.

Obviously F⁰(t) = h(t, N t)∈ R and thus F is a solution of the differential relation R. One crucial property of the convex integration process is that the solutionF can be made arbitrarily close to the initial map f₀..

Proposition (C⁰-density).– We have

kF −f₀k_C0 ≤ 1 N

2khk_C0 +k∂h

∂tk_C0

wherekgk_C0 = sup_p∈Dkg(p)k_Eⁿ denotes theC⁰ norm of a function g:D→ Eⁿ.

Proof.– Let t∈ [0,1]. We put n:= [N t] (the integer part of N t) and set I_j = [_N^j ,^j+1_N ] for 0≤j≤n−1 and I_n= [_Nⁿ, t].We write

F(t)−f(0) =

n

X

j=0

Sj and f0(t)−f0(0) =

n

X

j=0

sj

with Sj := R

Ijh(v, N v)dv and sj := R

Ij

R1

0 h(x, u)dudx. By the change of variables u=N v−j, we get for eachj∈[0, n−1]

S_j = 1 N

Z 1 0

h(u+j

N , u)du= Z

Ij

Z 1 0

h(u+j

N , u)dudx.

It ensues that

kS_j−s_jk

E³ ≤ 1 N²k∂h

∂tk_C0.

The proposition then follows from the obvious inequalities kS_n−s_nk_E3 ≤ 2

Nkhk_C0 and kF(t)−f₀(t)k_E3 ≤

n

X

j=0

kS_j−s_jk_E3.

The increase of theC⁰ closeness with N.

In a multi-variables setting, the convex integration formula take the following natural form:

f(c₁, ..., c_m) :=f₀(c₁, ..., cm−1,0) + Z cm

0

h(c₁, ..., cm−1, s, N s)ds

where (c₁, ..., c_m) ∈ [0,1]^m. This expression is nothing else but the para- metric formula of a convex integration process with parameter space P = [0,1]^m−1.It turns out that the above C⁰-density property can then be en- hanced to aC^1,mb-density property where the notationC^1,mb means that the closeness is measured with the following norm

kfk_C1,mb = max(kfk_C0,k∂f

∂c₁k_C0, ...,k ∂f

∂cm−1

k_C0),

that is theC¹-norm without thek ∂f

∂cm

k_C0 term.

Proposition (C^1,mb-density).– Let R ⊂ Rⁿ be an open set, E = C × Rⁿ−→^π C be the trivial bundle over the cube C = [0,1]^m, σ ∈Γ(R) and let f0:C −→Rⁿ be a map such that:

∀c= (c₁, ..., c_m)∈[0,1]^m, ∂f₀

∂c_m(c)∈ IntConv(R_c, σ(c))

where R_c =π⁻¹(c)∩ R. Then, for every > 0, there exists f :C −→ Rⁿ such that:

i) ∂f

∂cm

∈Γ(R) ii) ∂f

∂cm

is homotopic to σ in Γ(R) iii) kf −f₀k_C1,mb =O _N¹

.

Proof.– We have

∂f

∂cm

(c₁, ..., c_m) =h(c₁, ..., cm−1, c_m, N c_m)∈ R_c and _∂c^∂f

m(c1, ..., cm) is homotopic to σ(c) via

σu(c) :=hu(c1, ..., cm−1, cm, N cm)

whereh_u is the contracting map described just below the proof of the Fun- damental Lemma. Mimicking the proof of theC⁰-density property, it is easy to show that

k∂f

∂c_j −∂f₀

∂c_jk_C0 =O 1

N

for everyj∈ {1, ..., m−1}.

Remark.– Even if f₀(0) = f₀(1), the map F obtained by a convex inte- gration fromf0 does not satisfyF(0) =F(1) in general. This can be easily corrected by defining a new mapf with the formula

∀t∈[0,1] , f(t) =F(t)−t(F(1)−F(0)).

The following proposition shows that theC⁰-density property still holds for f and, provided N is large enough, that the map f is still a solution of R (which is assumed to be open).

Proposition.– We have

kf−f0k_C0 ≤ 2 N

2khk_C0+k∂h

∂tk_C0

andf⁰(R/Z)⊂ R.

Proof.– The first inequality is obvious. Indeed, from F(1)−F(0) =F(1)−f0(0) =F(1)−f0(1) we deduce

kf(t)−f₀(t)k ≤ kF(t)−f₀(t)k+kF(1)−f₀(1)k ≤2kF−f₀k_C0. Derivatingf we have f⁰(t) =F⁰(t)−(F(1)−F(0)) thus

kf⁰−F⁰k_C0 ≤ kF−f0k_C0 =O 1

N

.

SinceR is open, ifN is large enough f⁰(R/Z)⊂ R.

Remark.– It is of course easy to produce a parametric version of that proposition.

4 One dimensional h-principle

Definition.– A subset A⊂Rⁿ is ampleif for every a∈A the interior of the convex hull of the connected component to whichabelongs is Rⁿ i. e.: IntConv(A, a) =Rⁿ (in particularA=∅is ample).

A is not ample Ais ample A is not ample.

Example.– The complement of a linear subspaceF ⊂Rⁿ is ample if and only if CodimF ≥2.

Definition.– Let E=P ×Rⁿ−→^π P be a fiber bundle, a subsetR ⊂E is said to beampleif, for every p∈P,R_p:=π⁻¹(p)∩ R is ample inRⁿ. Remark.– If R ⊂ E is ample, then, for every p ∈ P, the condition z(p)∈Conv(R_p, σ(p)) necessarily holds.

Proposition.– Let E = R/Z×Rⁿ −→^π R/Z be a trivial bundle and let R ⊂E be an open and ample differential relation. Then, for everyσ ∈Γ(R), there existsf :R/Z−→Rⁿ such that

i) f⁰∈Γ(R),i. e. f ∈ Sol(R), ii) f⁰ is homotopic to σ in Γ(R).

Remark.– As a consequence, the natural

π₀(Sol(R))−→π₀(Γ(R)) is onto.

Proof.– Letf₀ :R/Z−→Rⁿ be a C¹ map. Since Ris ample, we have

∀t∈R/Z, f₀⁰(t)∈Rⁿ=IntConv(R_t, σ(t)).

IfN is large enough, the map f :R/Z−→Rⁿobtained fromf0 by a convex integration (with gluing)

∀t∈[0,1], f(t) :=f0(0) + Z t

0

h(s, N s)ds−t Z 1

0

h(s, N s)ds

is a solution of R. Thus, the point i. For all u ∈ [0,1], we define f_u : [0,1]−→Rⁿ by

∀t∈[0,1], fu(t) :=f0(0) + Z t

0

hu(s, N s)ds−u.t Z 1

0

h(s, N s)ds

whereh_u :R/Z×[0,1]−→ R is the natural deformation retract h_u(t, s) =

h(t, s) if s∈[0,^u₂]∪[1−^u₂] h(t, u) if s∈[^u₂,1− ^u₂].

Of courseh₁(t, s) =h(t, s) andh₀(t, s) =σ(t).The mapf_udoes not descend to the quotientR/Z= [0,1]/∂[0,1].But its derivative

f_u⁰(t) =h_u(t, N t)−u Z 1

0

h(s, N s)ds

induces a map fromR/ZinRⁿ since f_u⁰(0) =hu(0,0)−

Z 1 0

hu(s, N s)ds=σ(0)−u Z 1

0

h(s, N s)ds

f_u⁰(1) =hu(1, N)− Z 1

0

hu(s, N s)ds=σ(1)−u Z 1

0

h(s, N s)ds

and thusf_u⁰(0) =f_u⁰(1) becauseσ(0) =σ(1).Hence,σ_u :=f_u⁰ is a homotopy joiningf⁰=f₁⁰ toσ.Since

Z 1 0

h(s, N s)ds

=kF(1)−f₀(1)k=O 1

N

for every u ∈ [0,1] and t ∈ R/Z, the point σu(t) is as close as desired to h_u(t, N t)∈ R. Since R is open, it exists N such that, for all u ∈[0,1],we

haveσ_u∈Γ(R).This shows the pointii.

A parametric version of that proof allows to obtain the following theorem:

Theorem (One-dimensionalh-principle).–LetE =R/Z×Rⁿ−→^π R/Z be a un trivial bundle and let R ⊂ E be a open and ample differential relation, then the map

J :Sol(R)−→Γ(R) is a weak homotopy equivalence.

Observation.– Obviously, in the above theorem,R/Zcan be replaced by an interval.

5 Two applications of one-dimensional convex in- tegration

5.1 Whitney-Graustein Theorem

Whitney-Graustein Theorem (1937). – We have : π₀(I(S¹,R²))'Z, with an identification given by the tangential degree.

Proof.– The theorem is a direct application of the 1-dimensionalh-principle with n = 2 and R = R/Z× R²\ {(0,0)}

which is open and ample. We then have

Sol(R) =I(S¹,R²), Γ(R) =C⁰(R/Z,R²\ {(0,0)}) and

J : Sol(R) −→ Γ(R)

γ 7−→ γ⁰

induces a bijection at theπ0-level. Note that the components ofC⁰(R/Z,R²\ {(0,0)}) are in one to one correspondance withZ,the bijection being given by the turning number. It ensues thatπ₀(J) is the tangential degree.

5.2 A theorem of Ghomi

Theorem (Ghomi 2007).– Letf0 ∈I(R/Z,R³) be a curve with curvature function k₀ and let c be a real number such that c > max k₀. Then, for every >0, there exists f₁ ∈ I(R/Z,R³) of constant curvature c and such that

kf₁−f₀k_C1 =kf₁−f₀k_C0+kf₁⁰ −f₀⁰k_C0 ≤.

An example.– How toC¹ approximate a line by curve with an arbitrarily large constant curvature ? The answer lies in an picture :

Just a little comment however (from [4]: let us parametrize the line as a vertical segment in the three dimensional Euclidean space

f0(t) =



 0 0 t





with t ∈ [0,1]. The theorem asserts that there exists a curve with con- stant curvature c which is C¹-close to f0. A starting point is to begin by approximating the segment with an helix, for instance:

f₁(t) =





cosαt sinαt

t





whereα >0 and >0. The C⁰ closeness off₁ tof₀ is ruled by. Regard- ing the curvature, it is constant and can be made as large as we want by

decreasing α. However, as the number α is becoming large, the derivative moves far away from the derivative of the initial function. It ensues that the helix is notC¹ close tof₀.To correct that point, we need to reduce the horizontal variations of the function. Letk >0 and τ be two numbers, we set

f_k,τ(t) =















 k

k²+τ² cosp

k²+τ² t k

k²+τ² sinp

k²+τ² t

√ τ

k²+τ² t















 .

This is an helix with constant curvaturekand constant torsionτ. It is then visible that we have to choose a torsion notably bigger to the curvature to ensure a quasi-vertical derivative.

Skecth of the proof.– This is a good example of use of the 1-dimensional convex integration even if it is not an direct application of the 1-dimensional h-principle theorem. Here are the main steps:

1) First, reduce the problem to the case where the parametrization of f0 is given by the arc-length.Then, the curvature is the norm of the second derivative, that is the speed ofT0:=f₀⁰ :R/Z−→S².

2) Find T1 :R/Z−→S² with constant speed (in order to have a con- stant curvature) which isC⁰-close toT₀(to ensure thatkf₁⁰−f₀⁰k_C0 is small) and close in average toT₀ ( to get a small normkf₁−f₀k_C0).

3) Technically, T₁ should complete small loops with constant speed in a neighborhood of T0(R/Z) in S² and such that the average on each loop is close to the one ofT0 in the corresponding interval.

For more details, see [1].

References

[1] M. Ghomi, h-principles for curves and knots of constant curvature, Geom. Dedicata 127, 19-35, 2007. http://tinyurl.com/399y78f

[2] M. Gromov, Convex integration of differential relations I, Izv. Akad.

Nauk SSSR 37 (1973), 329-343.

[3] M. Gromov,Partial differential relations, Springer-Verlag, New York, 1986.

[4] M. Kourganov h-principe pour les courbes `a courbure con- stante, rapport de stage ENS-Lyon, http://math.univ-lyon1.fr/ bor- relli/Jeunes.html

[5] D. Spring, Convex Integration Theory, Monographs in Mathematics, Vol. 92, Birkh¨auser Verlag, 1998.