Regularity

In this paragraph we shall be concerned with derivatives of theF^ec,δ’s wherever this makes sense. Notice that X^e has a Riemannian manifold structure outside the set S_x0 = p⁻¹(x0). Hence, it makes sense to consider derivatives outside this set. Notice that, since we are dealing with a singular space, we have to face some difficulties concerning the regularity of the distance function. In this section we shall explain how this issue can be circumvented. We start proving the following lemma.

Lemma 6.12 (Regularity). The function B_c,y^δ (x) is C² onX \ S^e x0.

Since Bc,y^δ is obtained by integrating (˜ρδ,x)², Lemma 6.12 is a consequence of the next Lemma, concerning the regularity of the function ˜ρδ,z = ˜dδ(z, ·), and of Lebesgue’s Dominated Convergence Theorem.

Lemma 6.13. Let z ∈ X^e be any point. The function ˜ρδ,z : X −→ R^{e +} is C² on X r Se x0.

Proof. Let x ∈ X r S^e x0. Assume that z ∈ Y^e_i[g]i and x ∈ Y^e_j[g⁰]j. Since Y^e_i[g]i, Yej[g⁰]j are convex subsets of (X, ˜^e dδ) we can define Pi,g, Pj,g⁰ as the projections on these subsets. By construction it is readily seen that Pi,g(x) = (g · hi).˜x0 and P_j,g⁰(z) = (g⁰· h_j).˜x0, for suitably chosen hi, hj (depending only on Y^e_i[g]i,Y^e_j[g⁰]j).

Then,

˜ρδ,z(x) = ˜dδ(x, z) = ˜dδ(z, Pi,g(x)) + ˜dδ(Pi,g(x), Pj,g⁰(z)) + ˜dδ(Pj,g⁰(z), x).

Notice that, once z has been fixed, the first summand ˜d_δ(z, Pi,g(x)) and the second summand ˜dδ(Pi,g(x), Pj,g⁰(z)) are constant as long as x varies in Y^ej[g⁰]j. Moreover, by construction, the restriction of ˜d_δ toY^ei[g]i is equal to the Riemannian distance

function on (Y^ei,˜g_δⁱ) (and similarly the restriction of ˜d_δtoY^ej[g⁰]j). Hence, there exists a constant K = K(z, g, g⁰) such that

˜ρδ,z(x) = K + ˜d_δ(x, Pj,g⁰(z)) (6.5) for every point x ∈Y^ej[g⁰]j. By this remark, it is clear that we can take derivatives up to the second order (recall that the Riemannian metrics ˜gⁱ_δ’s are C²) of ˜ρδ,z at every point x ∈X r S^e x0, using the Levi-Civita connection with respect to the metric

˜g_δ^j of the appropriate leafY^e_j[g⁰]j to which x belongs.

Since we arleady know that for every y ∈Y^e the minimum of Bc,y^δ is attained at unique point, i.e. the barycentre F^e_c,δ(y) of the measure ˜f∗µ_c,y, Lemma 6.12 allows us to give the following characterization of F^e_c,δ(y) when F^e_c,δ(y) ∈X r S^e x0. The proofs of the following Lemmata 6.14, 6.15 and 6.16 follow exactly the same line as those of Lemmata 5.6, 5.7 and 5.8 in Chapter 5.

Lemma 6.14 (The implicit equation at manifold points). For every δ > 0, c >

Ent(g) and every y ∈Y^e such that x =F^ec,δ(y) ∈X r S^e x0, the point x is characterized by the equation:

Z

Xe

˜ρδ,z(x)d(˜ρδ,z)x(v)d( ˜f∗µc,y)(z) = 0, ∀v ∈ TxXe (6.6)

In other words, the map F^e_c,δ is univocally determined on the complement of the preimage of the singular set Sx0 by the vectorial implicit equation Gc,δ(F^e_c,δ(y), y) = 0, where Gc,δ = (Gⁱ_c,δ) : (X r S^e x0) ×Y −→ R^{e n} is defined as

Gⁱ_c,δ(x, y) = 1 2 Z

eY

d_x[(˜ρ_{δ, ˜f (y}⁰₎]²(Ei)e^{−c ˜d(y,y0)}dv_˜g(y⁰),

where {Ei}ⁿ_i=1 is a global ˜g^j_δ-orthonormal basis for the tangent space TxXe where x ∈Y^e_j[g⁰]j for some j, and ρ_{δ, ˜f (y}⁰₎(·) = ˜d_δ( ˜f(y⁰), ·).

Since x ∈X r S^e x0 we are allowed to take derivatives (up to the first order) of the function Gc(x, y). Hence, for those y ∈Y^e such that x =F^e_c,δ(y) ∈X r S^e x0 we have:

Lemma 6.15 (Derivatives of the implicit function Gc,δ). Using the notation intro-duced above, the map Gc,δ: (X rS^e x0)×Y → R^{e n} is C¹ and for x =F^e_c,δ(y) ∈X rS^e x0, the differential (dGc,δ)(x,y)|

Tx^Xe is non-singular. Hence, for x =F^e_c,δ(y) ∈X r S^e x0

the map F^ec,δ:Y →^e X^e is C¹ and satisfies:

d_yF^e_c,δ= − (dGc,δ)_(x,y)|⁻¹

Tx^Xe

◦(dGc,δ)_(x,y)|

Ty^Ye . Furthermore, if u ∈ TyYe and v ∈ TxXe we have:

(dGⁱc,δ)(x,y)(u) = −cZ eY

˜ρδ, ˜f (y⁰)(x) · dx(˜ρδ, ˜f (y⁰)(Ei)) · dy˜ρy⁰(u)e^{−c ˜d(y,y0)}dv˜g(y⁰)

(dGⁱc)(x,y)(v) = 1 2 Z

eY

Dd_x[˜ρδ, ˜f (y⁰)]²(v, Ei)e^{−c ˜d(y,y0)}dv_g˜(y⁰)

6.5 The Hessian of ˜ρδ,z on eX r Sx0 65

Finally, we have the following estimate of the Jacobian of the map F^e_c,δ at those points y ∈Y^e such that F^e_c,δ(y) ∈X r S^e x0:

Lemma 6.16 (Computation of the Jacobian ofF^e_c,δ ). Let us fix y ∈Y^e such that x=F^e_c,δ(y) ∈X r S^e x0 and define

ν_c,y^δ (y⁰) = ˜dδ( ˜f(y), ˜f(y⁰)) · µc,y(y⁰).

Furthermore, let us introduce the positive quadratic forms k_c,δ,y^X , h^X_c,δ,y and h^Y_c,δ,y defined on TxXe, TxXe and TyYe respectively.

k_c,δ,y^X (v, v) = _νδ ¹

c,y(^{Y )}e R

Ye

1

2Ddx[˜ρ_{δ, ˜f (y}⁰₎]²(v, v)dµc,y(y⁰), ∀v ∈ T_xXe h^X_c,δ,y(v, v) = ¹

ν_c,y^δ (^{Y )}e R

Yed_x[˜ρ_{δ, ˜f (y}⁰₎]²(v, v)dνc,y^δ (y⁰), ∀v ∈ T_xXe h^Y_c,δ,y(u, u) = _νδ¹

c,y(^{Y )}e R

Yed_y[(˜ρ)y⁰]²(u, u)dνc,y(y⁰), ∀u ∈ T_yYe

and let us denote by K_c,δ,y^X , H_c,δ,y^X and H_c,δ,y^Y the corresponding endomorphisms of the tangent spaces TxXe, TxXe and TyYe respectively, and associated to the scalar products (˜g_δ^j)x on TxXe = TxYej[g⁰]j (where we are assuming x ∈Y^ej[g⁰]j) and (˜g)y on T_yY^e. Then,

TrδH_c,δ,y^X = 1, Tr˜g(Hc,δ,y^Y ) = 1 (6.7) and the eigenvalues of H_c,δy^X and H_c,δ,y^Y are pinched between 0 and 1. Notice that we denoted for simplicity Trδ the trace with respect to the Riemannian metric on X r Se x0 given by the restriction of dδ.

Moreover, for any u ∈ TyYe and for any v ∈ TxXe the following inequality between the quadratic forms holds:

k^X_c,δ,y(dyFe_c,δ(u), v) ≤ c · h^X_c,δ,y(v, v) · h^Y_c,δ,y(u, u), (6.8) and

|Jacy(F^e_c,δ)| ≤ cⁿ

nⁿ² ·(det H_c,δ,y^X )¹²

det K_c,δ,y^X . (6.9)

In the next paragraph we shall obtain an estimate for the Hessian of ˜d_δ outside the singular set, which together with Lemma 6.16 will give the lower bound of the Minimal Entropy in the reducible case.

6.5 The Hessian of ˜ ρ

_δ,z

on X r S

^f x₀

By Lemma 6.16, in order to estimate JacF^e_c,δ we need to provide a good upper bound for the quadratic form k_c,δ,y^X . To this purpose we shall need an estimate of the Hessian of ˜ρδ,z where z ∈X r S^e x0. This will be done by looking at Hessx(˜ρδ,z) as the second fundamental form IIz,R(x) of the distance sphere Sδ(z, R) with center z and radius R = ˜d_δ(z, x), computed at x.

Lemma 6.17 (Hessian of ˜ρδ,z). Let us fix a point x ∈X r S^e x0 and a point z ∈X.^e Then IIz,R(x) = Hessx(˜ρδ,z).

Proof. Since x ∈ X r S^e x0 we know that there exists a unique i ∈ {1, .., k} and a unique [g]i ∈ G/G_i such that x ∈ Y^e_i[g]i. By construction of ˜d_δ the intersection of the geodesic sphere Sδ(z, R) passing through x with the leaf Y^ei[g]i is equal to S_δ(Pi,g(z), R − r) where r = ˜d_δ(Pi,g(z), z). But Sδ(Pi,g(z), R − r) is isometric to a suitable ˜g_δⁱ–geodesic sphere of the same radius in (Y^e_i,˜g_δⁱ). Thus:

IIz,R(x) = IIPi,g(z),R−r(x) = Hessx(˜ρⁱ_δ,Pi,g(z)) = Hessx(˜ρδ,z),

where in the second equality we denoted by ˜ρⁱ_δ,Pi,g(z) the Riemannian distance function ˜d_δ|

Yei[g]i from the point Pi,g(z); the third equality comes from Equation 6.5.

6.6 An approximated version of Rauch’s Comparison Theorem

From now on, we proceed following the same lines of the proof of the irreducible case, paying some extra care due to the presence of the singular points. Recall that we denoted yi ∈ Y_i for i = 1, . . . , k the points identified to the singular point x0 ∈ X. Proposition 6.18. For every ε > 0 define Rε= ln^q2_ε^. Assume that x ∈X r S^e x0

is such that, for any i = 1, . . . , k and g ∈ π1(X, x0), the intersection B

Xe(x, Rε)∩Y^ei[g]i

is either empty or is a hyperbolic ball; then for every z ∈ X^e and v ∈ TxX,e the following inequality holds:

(1 − ε)˜ρδ,z

||v||²_δ−(dx˜ρδ,z⊗ d_x˜ρδ,z)(v, v)^≤ ˜ρδ,zD_xd˜ρδ,z(v, v). (6.10)

Proof. We may assume that x ∈Y^ei[g]i. Recall that we denoted by Pg,i the projection onto the convex subspace Y^e_i[g]i. In order to prove Proposition 6.18 we need to distinguish two cases: the case where Pi,g(z) ∈ B

Yei[g]i(x, Rε) and the case where d˜δ(Pi,g(z), x) > Rε. In the first case, observe that the second fundamental form IIPi,g(z),r(x) of the ball of radius r = ˜d_δ,i(x, Pi,g(z)) —where we denote with ˜d_δ,i the Riemannian distance function on Y^ei[g]i— coincides with the second fundamental form in the hyperbolic case. Hence, for v ∈ TxXe

Hess_x(˜ρδ,z)(v, v) = Hessx(˜ρⁱ_δ,Pi,g(z))(v, v) ≥ ˜gδⁱ(v, v) − (dx˜ρⁱ_Pi,g(z)(v))², where we used the notation ˜ρⁱ_δ,w for the Riemannian distance function from w in (Y^ei[g]i,˜g_δⁱ). Otherwise, the distance between Pi,g(z) and x is greater than Rε, and

we can apply Lemma 5.12 in restriction toY^e_i[g]i, thus obtaining:

Hess_x(˜ρδ,z)(v, v) ≥ (1 − ε)^˜gδⁱ(v, v) − (dx˜ρδ,z(v))^2.

6.7 The Jacobian estimate 67

6.7 The Jacobian estimate

We are now ready to apply the inequalities found in Lemma 6.16 and Proposition 6.18, to derive an almost optimal inequality for the Jacobian of the mapsF^e_c,δ valid on an appropriately chosen subset of Y .

Proposition 6.19(Jacobian Estimate, reducible case). For every ε > 0 there exists R_ε>0 such that, if the ball centred at the non-singular point x =F^e_c,δ(y) ∈Y^e_i[g]i⊂Xe of radius Rε is such that B

Xe(x, Rε) ∩Y^e_i[gi] is hyperbolic for every g ∈ π1(X, x0), and every i = 1, ..., k, the following estimate for the Jacobian holds:

|JacF^e_c,δ(y)| ≤ 1

Xe(x, Rε) satisfies the assumptions. Then by Proposition 6.18 we know that, at the point x =F^e_c,δ(y) the following inequality holds:

and hence we obtain the inequality between the symmetric endomorphisms:

(1 − ε) ·^Id −Hc,δ,y^X

≤ K_c,δ,y^X

Now, using this inequality, Lemma 6.16 and Lemma 5.9 we obtain:

|JacF^e_c,δ(y)| ≤ cⁿ

The proof of Theorem 6.1 partly relies on the proof of Theorem 5.1. For every i= 1, ..., k and every j = 1, ..., ni we shall denote by {V_i,δ,ε^j }ⁿ_j=1ⁱ the collection of open subsets of (Yi, g_δⁱ) constructed in the proof of Theorem 6.1, when Yi does not admit a complete hyperbolic metric. Otherwise we shall choose V_i,δ,ε¹ = Yi. By construction of the metrics g_δⁱ (where g_δⁱ is the —unique— hyperbolic metric for every Yi which is of hyperbolic type) and by Proposition 6.19 we have that:

|Jac Fc,δ(y)| ≤ 1 (1 − ε)³ ·

c 2

3

for every y ∈ ^Ski=1

where the first equality comes from the coarea formula and the second from the fact that the V_i,δ,ε^j ’s are disjoint. Plugging the inequality for the Jacobian into the formula, we see that:

Now for every fixed ε > 0 the left-hand side of the latter inequality converges to Pk

As Inequality (6.12) holds for every Riemannian metric g on Y we have the following estimate from below for MinEnt(Y ):

MinEnt(Y ) ≥ 2 ·

69

Appendix A

Smooth and Riemannian orbifolds

This appendix is devoted to the introduction of smooth and Riemannian orbifolds.

For further details, we recommend [BMP03] and [KL14].

A.0.1 Definition and basic notions

Definition A.1. A smooth n-orbifold is a metrizable topological space O endowed with a collection^nUi, ˜Ui, φi,Γi

o

i, called an atlas, where for every i Ui is an open subset of O, ˜Ui is an open subset of Rⁿ⁻¹×[0, ∞), φi : ˜Ui → U_i is a continuous map (called a chart) and Γi is a finite group of diffeomorphisms of ˜Ui satisfying the following conditions:

1. The Ui’s cover O.

2. Each φi factors through a homeomorphism between ˜U_i/Γi and Ui.

3. The charts are compatible in the following sense: for every x ∈ ˜U_i and y ∈ ˜U_j with φi(x) = φj(y), there is a diffeomorphism ψ between a neighbourhood V of x and a neighbourhood W of y such that φj(ψ(z)) = φi(z) for all z ∈ V . For convenience, we shall assume that the atlas is maximal.

We remark that an orbifold O is a manifold if all the groups Γi are trivial. Given an orbifold O, we denote by |O| its underlying space, i.e. the topological space obtained by neglecting the orbifold structure.

Let x ∈ O be any point; then the local group of O at x is the group Γx defined as follows: pick any chart (U, ˜Ui, φ,Γ), where x ∈ U. Then Γx is the stabilizer of any point of φ⁻¹(x) under the action of Γ. It turns out that the group Γx is well defined up to isomorphism. We say that the point x is regular if Γx is trivial, otherwise x is singular. Notice that the set of singular points is empty if and only if O is a manifold.

There are several notions that can be defined in the orbifold context, extending in a straightforward manner the analogous notion given for manifolds.

The boundary of O, denoted ∂O, is the set of points x ∈ O such that there exists a chart at x, φi : ˜U_i → U_i such that φ⁻¹_i (x) ⊂ Rⁿ⁻¹× {0}. The orbifold O \ ∂O is called the interior of O. We call O orientable if it has an atlas such that each φi

and every element of each Γi are orientation preserving.

Observation. Let O be an orbifold with boundary, and assume that all the singular points are conical points, i.e. the group Γi' Zkacts by rotation. Let x be a boundary point. Then, there exists a chart ( ˜Ui, Ui.φi,Γi) at x such that φ⁻¹i (x) ⊂ Rⁿ⁻¹× {0}, and the group Γi is trivial. Thus, there is a neighbourhood of the boundary such that the local groups are trivial.

A.0.2 Maps between orbifolds

A map between two orbifolds O and O⁰ is a continuous map f : |O| → |O⁰| such that for every x ∈ O there are charts φi : ˜Ui → U_i3 x and φ⁰j : ˜U_j⁰ → U_j⁰ such that f(Ui) ⊂ U_j⁰ and the restriction f|Ui can be lifted to a smooth map ˜f : ˜Ui→ ˜U_j⁰ which is equivariant with respect to some homomorphism Γi →Γ⁰j. A map f : O → O⁰ is an immersion (resp. a submersion) if the lifts ˜f are immersions (resp. submersions).

An embedding is an immersion whose underlying map is a homeomorphism with the image. A diffeomorphism is a surjective embedding.

A.0.3 Orbifold covering

A covering of an orbifold O is an orbifold ˜O with a continuous map p : | ˜O| → |O|, called a covering map, such that every point x ∈ O has a neighbourhood U with the following property: for each component V of p⁻¹(U) there is a chart φ : ˜V → V such that p ◦ φ is a chart.

We say that an orbifold is good if it is covered by a manifold, bad otherwise. If an orbifold is finitely covered by a manifold we say that it is very good.

The notions of deck transformation group and universal covering extend to the context of orbifolds, see [Thu97] and [BMP03].

The fundamental group of O, denoted by π₁^orb(O), is the deck transformation group of its universal covering.

There is an interpretation of the fundamental group in terms of loops, keeping track of the passage through the singular locus. Furthermore, a continuous map between two orbifolds gives rise to a map between the corresponding orbifold fundamental groups.

A.0.4 Riemannian orbifolds

A Riemannin metric on an orbifold O is given by an atlas for O along with a collection of Riemannian metric on the ˜U_i’s so that:

1. Γi acts isometrically on ˜Ui for every i;

2. The diffeomorphism ψ is an isometry.

71

We say that a map f : O → O⁰ between Riemannian orbifolds is a Riemannian submersion (resp. a Riemannian immersion) if the lift ˜f is a Riemannian submersion (resp. a Riemannian immersion) equivariant with respect to the isometric actions of

the local groups.

Remark A.2. Let O and O⁰ be two orbifolds that are global quotients of two Riemannian manifolds M and M⁰ with respect to groups of isometries Γ and Γ⁰ acting properly. Then, a Riemannian submersion between O and O⁰ is equivalent to a Riemannian submersion between M and M⁰ which is equivariant with respect to the action of Γ and Γ⁰.

73

Appendix B

Proof of Proposition 3.12

Lemma B.1. Let Φ ∈ C¹(−∞, 0] ∩ C¹[0, +∞) ∪ C⁰(R) such that:

i. Φ⁰(t) ≥ 0 for every t 6= 0;

ii. |Φ(0)| ≤ M;

iii. 0 ≤ Φ⁰(0)⁻<Φ⁰(0)⁺

Then there exists ε > 0, ε arbitrarily small, such that there exists a function Φε∈ C¹(R) satisfying the following properties:

a. Φε(t) = Φ(t) for t ∈ R \ (−ε, ε) and Φ ≤ Φε≤Φ(ε) for t ∈ (−ε, ε) b. Φ⁰(−ε) ≤ Φ⁰ε(t) ≤ Φ⁰(ε) for every t ∈ (−ε, ε)

c. |^R_−ε^ε Φε(s)ds| ≤ 2Mε

Proof. Property iii implies that there exists ε1 such that, for ε < ε1, inequality Φ⁰(−ε) < Φ⁰(ε) holds. Let us choose ε < ε1; then it si possible to construct a function Φε such that Φε(±ε) = Φ(±ε), Φ⁰_ε(±ε) = Φ⁰(±ε) and Φ⁰(−ε) ≤ Φ⁰(t) ≤ Φ⁰(ε).

Furthermore, we can choose Φε such that Φ(−ε) ≤ Φε(t) ≤ Φ(ε). Moreover, since

|Φ(0)| < M there exists ε2 such that for ε < ε2 inequality |Φ(t)| ≤ M holds for t ∈(−ε, ε). Thus, provided ε < min(ε1, ε₂):

   

Z ε

−εΦε(s)ds 

≤ Z ε

−ε

|Φε(s)|ds ≤ 2Mε

Lemma B.2. Let ϕ ∈ C²(−∞, 0] ∩ C²[0, +∞) ∩ C¹(R) with ϕ(0) ≥ 0 such that:

i. ϕ convex;

ii. |ϕ⁰(0)| ≤ M;

iii. 0 ≤ ϕ⁰⁰(0)⁻< ϕ⁰⁰(0)⁺

Then there exists ε > 0 arbitrarily small and ϕε∈ C²(R) such that:

A. ϕ⁰(t) ≤ ϕ⁰ε(t) ≤ ϕ⁰(ε) for every t ∈ (−ε, ε) ;

B. ϕ⁰⁰(−ε) ≤ ϕ⁰⁰ε(t) ≤ ϕ⁰⁰(ε) for t ∈ (−ε, ε);

Proof. Denote by Φ = ϕ⁰. Then, Φ satisfies the hypotheses of Lemma B.1 and applying it we get a function Φε such that properties a,b,c of Lemma B.1 hold. Let us define:

ϕε(t) =^{Z t}

−εΦε(s)ds + ϕ(−ε) We show that ϕε satisfies property C.

For t ≤ −ε,

For t ∈ (−ε, ε), the same computation just held gives:

|ϕ_ε(t) − ϕ(t)| ≤ 

Z t

−εΦε(s)ds

+ |ϕ(−ε) − ϕ(t)| ≤ 4Mε

Finally, we observe that property a of Lemma B.1 implies the thesis A, and part B of the thesis follows from property b.

Proposition B.3. For every fixed `, δ > 0 there exists ε > 0 arbitrarily small and ϕ_ε∈ C²(R) such that:

1. ϕε is not increasing and convex, such that ( ϕ_ε(t) = ϕ0(t) = `e^−t for t ≤ −ε

75

Proof. The function ϕ0(t) = `e^−t satisfies the differential equation ^ϕ_ϕ⁰⁰⁰ = 1, with

initial conditions ₍

Straightforward computations show that, for δ > 0, the function ϕδ is more convex than ϕ0, and ϕδ has a unique stationary point, which turns to be a minimum, at tδ= _2(1+2δ)¹ ln^1 +¹_δ^. We denote by `δ the corresponding minimal value. C²(R). Furthermore, ϕ is not increasing, convex and satisfies condition (1). Moreover, we observe that, for every δ > 0:

ϕ⁰²_δ ϕ²_δ ≤ ϕ⁰²₀

ϕ²₀. (B.1)

Indeed, on the interval [0, tδ] the following inequality holds:

ϕ⁰⁰_δ

The latter Inequality implies Equation B.1. In order to obtain a C²function satisfying the required properties we will apply (twice) Lemma B.2, as follows.

Firstly, we observe that ϕ satisfies the hypotheses of Lemma B.2; hence, there exists a function ϕε, enjoying properties (A), (B), (C). In particular, ϕε is convex, not increasing, and

We claim that ϕε satisfies properties 2) and 3). Indeed, Lemma B.2 implies:

where the latter inequality holds provided that δ < ¹₂. Thus, ϕε satisfies Property 2.

We claim that ϕε also satisfies Property 3. Lemma B.2 implies:

ϕ⁰⁰(−ε) ≤ ϕ⁰⁰ε(t) ≤ ϕ⁰⁰(ε) where the latter inequality holds provided that δ ≤ ¹₂.

Let us define ψε(t) = ϕε(tδ− t). We observe that ψε(0) = ϕε(tδ), ψε(ε) = ϕε(tδ− ε) and ψε(−ε) = ϕε(tδ+ ε). Furthermore, ψε satisfies the hypothesis of Lemma B.2.

Then there exists ε⁰ sufficiently small, and a convex function ψε,ε⁰(t) such that:



77

ψ⁰⁰_ε(−ε⁰) ≤ ψε,ε⁰(t) ≤ ψε⁰⁰(ε⁰) ⇐⇒ ϕ⁰⁰ε(t + ε⁰) ≤ ψε,ε^{00 0}(t) ≤ ϕ⁰⁰ε(tδ− t) (B.3) In order to prove that ψε,ε⁰ satisfies Property 2, we perform the following estimates, exploiting Equation B.2:

where the latter inequality holds provided that we have chosen ε sufficiently small so that |cε| ≤ ^δϕ₁₀₀^δ(tδ).

where the second inequality holds if we choose ε⁰ small enough so that

−ϕ⁰_δ(tδ− ε⁰) < 1 2ϕ(tδ)

Finally, we check that ψε,ε⁰ satisfies Property 3. Exploiting Equation B.3 we obtain:

ψ⁰⁰_ε,ε0(t)

ψε,ε⁰(t)≤ ϕ⁰⁰_ε(tδ− ε⁰)

ϕε(tδ− t) − 3Mε⁰ ≤ ϕ⁰⁰_ε(tδ− ε⁰) ϕ(tδ+ ε⁰) − 3Mε⁰− |cε| We choose ε⁰ sufficiently small so that

|c_ε|+ 3Mε⁰ ≤ δ

where the latter inequality holds provided that ε⁰ is sufficiently small so that ϕ(tδ− ε⁰)^1 − δ

By construction, this function satisfies the required properties.

79

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