MULTIFRACTAL PROPERTIES OF TYPICAL CONVEX FUNCTIONS

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MULTIFRACTAL PROPERTIES OF TYPICAL CONVEX FUNCTIONS

Zoltan Buczolich, Stéphane Seuret

To cite this version:

Zoltan Buczolich, Stéphane Seuret. MULTIFRACTAL PROPERTIES OF TYPICAL CONVEX FUNCTIONS. Monatshefte für Mathematik, Springer Verlag, In press. �hal-01612290�

FUNCTIONS

ZOLT ´AN BUCZOLICH AND ST ´EPHANE SEURET

Abstract. We study the singularity (multifractal) spectrum of continuous convex functions defined on [0,1]^d. Let Ef(h) be the set of points at whichf has a pointwise exponent equal toh. We first obtain general upper bounds for the Hausdorff dimension of these setsEf(h), for all convex functionsf and allh≥0. We prove that for typical/generic (in the sense of Baire) continuous convex functions f : [0,1]^d → R, one has dimEf(h) = d−2 +hfor all h∈[1,2],and in addition, we obtain that the set Ef(h) is empty ifh∈ (0,1)∪(1,+∞). Also, when f is typical, the boundary of [0,1]^d belongs toEf(0).

1. Introduction and main results

In this paper we investigate the multifractal properties of the contin- uous convex functions defined on [0,1]^d. This paper is a quite natural continuation of our papers [3, 4] where generic multifractal properties of measures and of functions monotone increasing in several variables (in short: MISV) were studied. It is interesting that all these natu- ral objects supported on [0,1]^d have very different typical multifractal behaviors.

Let us first recall that the pointwise H¨older exponent and the singu- larity spectrum for a locally bounded function are defined as follows.

Definition 1. Let f ∈ L^∞([0,1]^d). For h ≥ 0 and x ∈ [0,1]^d, the function f belongs to C^h(x) if there are a polynomial P of degree strictly less than [h] and a constant C > 0 such that, for allx⁰ close to x,

(1) |f(x⁰)−P(x⁰−x)| ≤C|x⁰−x|^h.

Date: February 28, 2017.

Research supported by the Hungarian National Research, Development and In- novation Office–NKFIH, Grant 104178.

Research partly supported by the grant ANR MUTADIS ANR-11-JS01-0009.

1

The pointwise H¨older exponent of f atx is

h_f(x) = sup{h≥0 : f ∈C^h(x)}.

In the following, dim_H denotes the Hausdorff dimension.

Definition 2. The singularity spectrum of f is the mapping d_f(h) = dim_HE_f(h), whereE_f(h) ={x:h_f(x) = h}.

By convention dim∅=−∞. We will also use the sets (2) E_f^≤(h) ={x:hf(x)≤h} ⊃Ef(h).

We denote byCC^dthe set of continuous convex functionsf : [0,1]^d→ R. Equipped with the supremum normk·k,CC^dis a separable complete metric space. An open ball in CC^d of center f ∈ CC^d of radiusr ≥0 is written as Bk·k(f, r), and a closed ball is Bk·k(f, r)

In this paper, we first prove an upper bound for the multifractal spectrum of all functions in CC^d.

Theorem 1. For any function f ∈ CC^d, one has

d_f(h)≤







d−1 if h∈[0,1) d+h−2 if h∈[1,2]

2 if h >2.

Then we compute the multifractal spectrum of typical functions in CC^d. Recall that a property is typical, or generic in a complete metric space E, when it holds on a residual set, i.e. a set with a complement of first Baire category (a set is of first Baire category if it is the union of countably many nowhere dense sets).

Theorem 2. For typical functions f ∈ CC^d, one has

d_f(h) =







d−1 if h= 0 d+h−2 if h∈[1,2]

−∞ otherwise.

More precisely, one has E_f(0) =∂([0,1]^d).

It is interesting to compare Theorem 2 with the regularity of other typical objects naturally defined on the cube [0,1]^d.

When d = 1, generic properties of continuous functions on [0,1]

have been studied for a long time (see for instance [5, 6], and the other references we mention in the present paper). It was proved that typical continuous functions belong to C⁰(x), for every x ∈ [0,1]. In [2], typical monotone continuous functions on the interval [0,1] were

0 1 d−1

d

1 d

Generic MISV functions and measures in one dimension

Generic measures in Rd

Generic MISV functions in R Generic MISV functions in R

Generic convex functions in Rd Generic MISV functions in Rd

2

Figure 1. Typical spectra for measures, for MISV and for convex functions

proved to have a much more interesting multifractal behavior: such functions f : [0,1]→R satisfy

(3) d_f(h) = dimE_f^≤(h) =

( h if h∈[0,1]

−∞ otherwise.

The same holds true for typical monotone functions (not necessarily continuous). We remark that it also follows, for example from results in [2], that for arbitrary monotone functions there is an upper estimate (4) dimE_f^≤(h)≤h for h∈[0,1].

It is interesting to extend these results to higher dimensions.

The first natural way is to consider Borel measures on the cube. In [3] (see also [1] for a nice generalization to all compact sets in R^d), it is proved that typical measures µ supported on [0,1]^d satisfy a multi- fractal formalism, and

d_µ(h) =

( h if h∈[0, d]

−∞ otherwise.

Another interesting class is constituted of the continuous monotone increasing in several variables (in short: MISV) functions. These func- tions extend to higher dimensions in a different direction the one- dimensional monotone functions. A function f : [0,1]^d → R is MISV when for all i∈ {1, ..., d}, the functions

(5) f⁽ⁱ⁾(t) = f(x₁, ..., x_i−1, t, x_i+1, ..., x_d)

are continuous monotone increasing. We use the notation M^d ={f ∈C([0,1]^d) :f MISV}.

The space M^d is a separable complete metric space when equipped with the supremum L^∞ norm for functions. Typical MISV functions satisfy (see [2, 4])

(6) df(h) =

(d−1 +h if h∈[0,1]

−∞ otherwise.

On Figure 1 we compare our new results about generic continuous convex functions with the earlier results we mentioned.

Remark 1. One cannot directly infer Theorems 1 and 2 by integrating MISV functions or measures on [0,1]^d. For instance, lettingf(x₁, x₂) = 10(x²₁ +x²₂) +x²₁x²₂, its second differential d²f(x₁, x₂, h₁, h₂) = (20 + 2x²₂)h²₁+ (20 + 2x²₁)h²₂+ 4x₁x₂h₁h₂ is positive definite for any (x₁, x₂)∈ [−1,1]². Hence this function f is strictly convex on [−1,1]², but∂1f = 20x1+2x1x²₂is monotone only inx1and∂2f = 20x2+2x2x²₁is monotone only in x₂.

2. Preliminary Results

2.1. Basic notation. If it is not stated otherwise we work in R^d. Points in the space are denoted byx= (x₁, ..., x_d). Thej’th unitvector is denoted by

e_j = (0, ...,0,1

↑

j

,0, ...,0).

Open balls in R^d are denoted by B(x, r) (to be distinguished from the balls Bk·k(f, ε) in CC^d).

2.2. Local regularity results of continuous convex functions.

First, recall that C^∞ functions are dense in CC^d.

Remark 2. To see this, take ψ ≥0 a C^∞ function, which is 0 outside B(0,1), such that

Z

R^d

ψ = 1. Put ψ_λ = λ^dψ(¹_λx). If f is convex and continuous on [−1,1]^d, then the convolutionf_λ(x) =R

[0,1]^df(x)ψ_λ(y− x)dyis convex on [−1 +λ,1−λ]^d and f_λ(x) =f_λ(_1−λ¹ x) is convex on [−1,1]^d. Using the uniform continuity of f on [−1,1]^d, one easily sees that||f−f_λ|| →0 asλ→0.One concludes by using an obvious linear transformation.

A first lemma allows one to control the left and right partial deriva- tives of all functions in a neighborhood of a convex differentiable func- tion in CC^d. The notations ∂_j,+f and ∂_j,−f are used for the right and left partial j-th derivatives of a convex functionf.

Lemma 3. Suppose f ∈ CC^d∩ C¹([0,1]^d) and ε > 0. There exists

%_f,ε > 0, such that for all j ∈ {1, ..., d}, if g ∈ B_k·k(f, %_f,ε), then for every x_j ∈[ε,1−ε], x_i ∈[0,1], i∈ {1, ..., d} \ {j} we have

(7) |∂j,±g(x₁, ..., x_d)−∂_jf(x₁, ..., x_d)|< ε.

Proof. It is enough to fix one j ∈ {1, ..., d}. Recall that f ∈ CC^d∩ C¹([0,1]^d) implies that ∂_jf is non-decreasing in x_j and is uniformly continuous on [0,1]^d.

Using this uniform continuity, there exists a partition 0 = xj,0 <

xj,1 < ... < xj,K = 1 such that xj,1 < ε and for every xi ∈ [0,1] with i∈ {1, ..., d} \ {j}, one has for every l= 1, ..., K,

(8)

∂_jf(x₁, ..., xj−1, x_j,l, x_j+1, ..., x_d)−∂_jf(x₁, ..., xj−1, xj,l−1, x_j+1, ..., x_d)< ε 4. Set

(9) %f,ε,j = ε

4 min

l=2,...,K(xj,l−xj,l−1).

Consider any function g ∈ B_k·k(f, %_f,ε,j), and fix x = (x₁, ..., x_d) ∈ [0,1]^d, so that xj ∈[ε,1−ε].

There exists an integerl ∈ {1, ..., K−1} such thatx_j ∈[x_j,l, x_j,l+1].

Set x(l⁰) = (x₁, ..., x_j−1, x_j,l⁰, x_j+1, ..., x_d).

Since the two functions t 7→ g(x₁, ..., x_j−1, t, x_j+1, ..., x_d) and t 7→

f(x₁, ..., xj−1, t, x_j+1, ..., x_d) are real convex, one has

∂j,−g(x) ≥ ∂j,−g(x(l))≥ g(x(l))−g(x(l−1)) x_j,l−xj,l−1

≥ f(x(l))−f(x(l−1))−2%_f,ε,j x_j,l −xj,l−1

≥ f(x(l))−f(x(l−1)) x_j,l −x_j,l−1 − ε

2

≥ ∂_jf(x(l−1))− ε 2. (10)

Thanks to the convexity oft7→f(x₁, ..., xj−1, t, x_j+1, ..., x_d), equation (8) gives

∂_jf(x)≤∂_jf(x(l+ 1)) < ∂_jf(x(l−1)) + 2ε 4.

Hence, one can continue (10) to obtain

∂j,−g(x)> ∂_jf(x)− ε 2− ε

2 =∂_jf(x)−ε.

Moreover, a similar argument can show that

∂_j,+g(x)< ∂_jf(x) +ε.

This ends the proof, since ∂_j,−g(x)≤∂_j,+g(x).

The conclusion follows by taking %_f,ε = min(%_f,ε,j :j = 1, ..., d).

The one-dimensional version of Lemma 3 is stated as follows.

Lemma 4. Suppose f ∈ CC¹ ∩C¹([0,1]). For ε >0 there exist ε >0 and %_ε,f > 0 such that for any g ∈ Bk·k(f, %_ε,f) and x ∈ [ε,1−ε] we have

(11) |g⁰_±(x)−f⁰(x)|< ε.

Next, one compares the pointwise exponents of a differentiable con- vex function f and its derivative f⁰. It is a general property that hf⁰(x)≤hf(x)−1, for every differentiable f. A surprising property is that equality necessarily holds when f is convex and h_f(x)∈[1,2), Lemma 5. Iff is convex and differentiable on(a, b)andhf(x)∈[1,2) for some x∈(a, b), then hf(x) =hf⁰(x) + 1.

Proof. It is enough to prove that h_f(x) ≤ h_f⁰(x) + 1. Since h_f(x) ∈ [1,2), necessarily h = hf⁰(x) < 1. Hence, there exists a sequence (x_n)_n≥1 converging toxsuch that|f⁰(x_n)−f⁰(x)|>|x_n−x|^h+n1.With- out limiting generality, suppose that x_n > x. By the monotonicity of f⁰, one has

f⁰(x_n)> f⁰(x) + (x_n−x)^h+1n. By convexity of f,

f(x_n)≥f(x) +f⁰(x)(x_n−x).

Setting x⁰_n =x+ 2(x_n−x), and using again the convexity of f at x_n, one gets

f(x⁰_n) ≥ f(xn) +f⁰(xn)(x⁰_n−xn)

≥ f(x) +f⁰(x)(x_n−x) + (f⁰(x) + (x_n−x)^h+1n)(x⁰_n−x_n)

≥ f(x) +f⁰(x)(x⁰_n−x) + (x_n−x)^h+n11

2(x⁰_n−x)

= f(x) +f⁰(x)(x⁰_n−x) + 1

2^h+1+1n(x⁰_n−x)^h+1+n1.

This implies h_f(x)≤h+ 1, hence the result.

One investigates what happens for non-differentiable convex func- tions.

Lemma 6. If f is convex on (a, b) ⊂ R and hf(x) ∈ [1,2) for some x∈(a, b), then min(h_f+⁰ (x), h_f−⁰ (x))≤hf(x)−1.

Proof. When hf(x) = 1 the lemma is obvious. Set h=hf(x)>1, and let ε > 0 so that h−ε > 0. By definition, there exists M ∈ R such that one has

|f(x+y)−f(x)−M y| ≤ |y|^h−ε

for every small y, and there exists a sequence (y_n)n≥1 converging to zero such that

|f(x+y_n)−f(x)−M y_n| ≥ |y_n|^h+ε. Hence,

|y_n|^h+ε−1 ≤

f(x+yn)−f(x)

y_n −M

≤ |y_n|^h−ε−1.

Since the left and right derivatives f₊⁰ (x) and f₋⁰ (x) both exist, they both equal M =f⁰(x).

Assume, without loss of generality, that there are infinitely many positive y_n’s. For every y_n,

f₊⁰ (x+y_n)−f₊⁰ (x)≥ f(x+yn)−f(x)

y_n −f₊⁰(x)≥ |y_n|^h+ε−1, thus h_f⁰

+(x)≤h+ε−1, which gives the result.

We also prove the following proposition, which somehow asserts that a convex function cannot have exceptional isolated directional point- wise regularity.

For this, consider the d-dimensional unit sphere S_d = {x ∈ R^d : kxk = 1}. Then, we select a finite set of pairwise distinct points (z₁,z₂, ...z_N)∈(S_d)^N for some integerN ≥1 such that the convex hull of {z₁,z₂, ...z_N}contains the d-dimensional ballB(0,1/2).

Let us choose ε_c>0 so small that :

• 0< ε_c≤ ₁₀₀₀¹ min(kz_i−z_jk, i6=j, i, j ∈ {1, ..., N}).

• Setting for every i∈ {1, ..., N} (12) C_i =S_d∩B(z_i, ε_c),

then for any choice ofz⁰_i ∈C_i, the convex hull of{z⁰₁,z⁰₂, ...z⁰_N} contains the ball B(0,1/4).

Proposition 7. If hf(x) = h, then there exists i ∈ {1, ..., N} such that for every z⁰_i ∈ C_i (see (12)), the restriction of f to the straight line passing through x parallel to the vector z⁰_i has a pointwise H¨older exponent equal to h.

Proof. Let n be such that n ≤ h < n + 1. We assume without loss of generality that x = 0, f(0) = 0, D^kf(0, ...,0) = 0 for every k ∈ {1, ..., n}. Let ε > 0. By definition, for every x close to 0, |f(x)| ≤

|x|^h−ε, and there exists a sequence (xn = (xn,1, ..., xn,d))n≥1 of elements inR^d, converging to0, such that

(13) |x_n|^h+ε ≤ |f(x_n)| ≤ |x_n|^h−ε.

Consider such an elementx_n, and the sets (C_n,i := 4kx_nk ·C_i)_i=1,...,N. Let us prove that there existsi_n ∈ {1, ..., N}such that for every n∈N and z⁰_in ∈C_n,in, |f(z⁰_in)| ≥(|z⁰_in|/4)^h+ε.

Assume first that for everyi∈ {1, ..., N}, there exists z⁰_i ∈C_n,i such that|f(z⁰_i)|<|f(xn)|. By construction, the ballB(0,kxnk) is included in the convex hull of these points (z⁰_i)i=1,...,N. By convexity, this would imply that |f(x_n)| ≤max(|f(z⁰_i)|:i= 1, ..., N), hence a contradiction.

Hence, there exists ani_n ∈ {1, ..., N}such that for everyz⁰_i

n ∈C_n,in,

|f(z⁰_in)| ≥ |f(xn)| ≥ |xn|^h+ε = (|z⁰_in|/4)^h+ε.

Turning to a subsequence, one can assume thati_n is constant, equal toi∈ {1, ..., N}.

Now, as a consequence of what precedes, for every vector z⁰_i ∈ C_i, there exists an infinite number of values (r_n = |x_n|)n≥1 such that

|f(r_nz⁰_i)| ≥ (|r_nz⁰_i|/4)^h+ε. Hence, the restriction of f to the straight line passing through 0 parallel to z⁰_i has a pointwise H¨older exponent less than h+ε. Since this holds for every ε > 0, and obviously this exponent is bounded below byh(i.e. the exponent off), one concludes that this restriction has exactly exponent h.

3. First typical properties of continuous convex functions

Based on Lemma 3 it is very easy to see that the typical function in CC^d is continuously differentiable on (0,1)^d. This was proved in [5, 6]

for instance. We give another proof for completeness.

Proposition 8. There is a dense Gδ set G in CC^d such that every f ∈ G is continuously differentiable on(0,1)^d.

Proof. By convexity, the partial derivatives ∂j,±f(x) exist for any f ∈ CC^d, x∈(0,1)^d and j ∈ {1, ..., d}.

SinceCC^dis separable, one can choose a sequence of convex functions {f_m : m = 1, ...} dense in CC^d. In addition, by Remark 2, one can assume that all these functions f_m are C^∞([0,1]^d) functions.

By uniform continuity of all the partial derivatives of f_m, there is a δ_n,m >0 such that for everyj, for every x,x⁰ ∈[0,1]^d,

(14) |∂_jf_m(x)−∂_jf_m(x⁰)|< 1

n, when |x−x⁰|< δ_n,m.

Applying Lemma 3, it is possible to choose 0< %n,m < _n+m¹ such that if f ∈ Bk·k(f_m, %_n,m) then for every j ∈ {1, ..., d}, if x = (x₁, ..., x_d) ∈ [0,1]^d with x_j ∈[_n¹,1−_n¹], then

(15) |∂j,±f(x)−∂_jf_m(x)|< 1 n. Let us introduce the sets

G_n=

∞

[

m=1

Bk·k(f_m, %_n,m) and G=

∞

\

n=1

G_n. It is clear that G is a dense G_δ set in CC^d.

We prove that any f ∈ G is continuously differentiable.

By definition, there exists an infinite sequence of integers (m_n)n≥1

such that f ∈Bk·k(f_mn, %_n,mn).

Fixj ∈ {1, .., d}, and focus on the j-th partial derivatives. Combin- ing inequalities (14) and (15), if x,x⁰ ∈ [0,1]^d, with x_j, x⁰_j ∈ [_n¹,1− ¹_n] and ||x−x⁰||< %_n,mn, then

|∂j,±f(x)−∂j,±f(x⁰)|<|∂_jf_mn(x)−∂_jf_mn(x⁰)|+ 2 n < 3

n. The ± is the above inequality means that any choice of left or right derivative can be made.

From this, lettingn→ ∞, it follows easily that ∂j,+f(x) = ∂j,−f(x),

hence ∂_jf is continuous on (0,1)^d.

However, the next lemma shows that typical functions f inCC^d are not differentiable on [0,1]^d. The problem comes from the boundary of the domain.

Proposition 9. There is a dense G_δ set G∞ in G such that everyf ∈ G∞ satisfies the following: for every j ∈ {1, ..., d}, for every x_i ∈[0,1]

with i∈ {1, ..., d} \ {j}, one has

(16) ∂_j,+f(x₁, ..., xj−1,0, x_j+1, ..., x_d) = −∞

and

(17) ∂_j,−f(x₁, ..., x_j−1,1, x_j+1, ..., x_d) = +∞.

Moreover, (18)

h_f(x₁, ..., x_j−1,0, x_j+1, ..., x_d) = 0 =h_f(x₁, ..., x_j−1,1, x_j+1, ..., x_d).

Proof. We are going to show that for a fixedj, there is a denseG_δ set G^0,j inCC^d such that iff ∈ G^0,j then (16) and the first equality in (18) holds.

Without loss of generality, one considers j = 1. As in the proof of Theorem 8, one chooses a sequence of C^∞ functions (f_m)m≥1. We also select integer constants M_m ≥1 such that |∂₁f_m| ≤M_m on [0,1]^d.

For every integer l≥1, let us introduce the mapping ϕ_l defined as ϕ_l(x₁, ..., x_d) =

(−l^l−1x₁ if 0≤x₁ ≤l^−l,

−l⁻¹ if l^−l ≤x₁ ≤1.

Clearly,ϕl is convex and for any fixed nthe functionsfm+ϕn·m·Mm, m= 1, ...are dense in CC^d. Set

G_n^0,1 =

∞

[

m=1

Bk·k(f_m+ϕn·m·Mm,(n·m·M_m)^−n·m·Mm), and

G^0,1 =

∞

\

n=1

G_n^0,1.

We prove that if f ∈ G^0,1, then (18) and hence (16) hold. By defini- tion, there exists an infinite sequence of integers (m_n)n≥1 such that

f ∈Bk·k f_mn +ϕn·m_n·M_mn,(n·m_n·M_mn)^{−n·mn·Mmn} . For ease of notation, set l_n =n·m_n·M_mn, that is

f ∈Bk·k(f_mn +ϕ_ln, l^−l_n ⁿ).

Consider x∈[0,1]^d, with x1 = 0. Then for every n ≥5, f(0, x2, ..., xd)−f(l^−l_n ⁿ, x2, ..., xd)

≥ (f_mn +ϕ_ln)(0, x₂, ..., x_d))−(f_mn +ϕ_ln)(l^−l_n ⁿ, x₂, ..., x_d))−2·l^−l_n ⁿ

≥ −M_mnl^−l_n ⁿ+l⁻¹_n −2·l^−l_n ⁿ

= l⁻¹_n

1− M_mn

l_n l_n^−ln+2− 2

l_nl_n^−ln+1

> 1 2ln

,

where we have used the boundedness of∂fmn byMmn and the fact that l_n >> M_mn. Hence, for anyα >0 we have

f(0, x₂, ..., x_d)−f(l^−l_n ⁿ, x₂, ..., x_d) l^−l_n ^nα > 1

2l_n^lnα−1,

which tends to infinity when n goes to infinity. Hence (16) holds true, and one also deduces that h_f(0, x₂, ..., x_d) = 0.

Similar arguments yield Gδ sets G^0,j for j > 1 and G^1,j for j = 1, ..., dsuch that the functionsf in these sets satisfy the corresponding equalities in (16-18).

Finally, the set

G∞ =

d

\

j=1

G^0,j∩ G^1,j

satisfies the conditions of Proposition 9.

We conclude this section by proving that typical convex functions have only pointwise exponents less than 2.

Proposition 10. There exists a G_δ-set G² such that for everyf ∈ G², E_f(h) = ∅ for every h >2.

Before proving Proposition 10, we introduce some perturbation func- tions already used in Section 4 of [4].

Definition 3. For everyl∈N, the functionγ_l : [0,1]→[0,1] is defined as follows:

• γ_l is continuous,

• For every integer j = 0, ...,2^l2 −1, if x₁ ∈[j2^−l2,(j+ 1)2^−l2 − 2^−l4], one sets γ_l(x₁) = j2^−l2−l. So γ_l is constant on these intervals,

• γ_l(1) = 2^−l,

• For every integer j = 0, ...,2^l2 −1, the mapping γl is affine on the intervals [(j + 1)2^−l2 −2^−l4,(j+ 1)2^−l2],

These functions γ_l are continuous, ranging from 0 to 2^−l, and are strictly increasing only on the 2^l2 many very small, uniformly dis- tributed, intervals of length 2^−l4.

Proof. We start by selecting a set of C^∞ functions {f_m :m = 1,2, ...}

which is dense in CC^d. We choose an integer M_m,3 ≥ 1 such that the second and third partial derivatives ∂₁³fm with respect to the first variable x1 of fm satisfy |∂₁²fm|+|∂₁³fm| ≤Mm,3.

Observe that the x₁-partial derivatives ∂₁f_m of these functions are monotone in the first variable.

Then, one introduces the auxiliary functions, depending only on the first variable: for l≥0,

(19) γ_l(x₁, x₂, ..., x_d) =γ_l(x₁).

The perturbation functions are defined for l ≥0 by (20) f_l(x) = f_l(x₁, x₂, ..., x_d) =

Z x1

0

γ_l(t, x₂, ..., x_d)dt = Z x1

0

γ_l(t)dt.

Next we apply Lemma 3 to the functions f_m+f_m+Mm,3+n with ε= ε_m,n := 2^{−(m+Mm,3+n)8} and j = 1. There exists a constant, denoted by

%_m,n > 0, such that if f ∈ Bk·k(f_m +f_m+Mm,3+n, %_m,n), then for any x1 ∈[εm,n,1−εm,n] and xj ∈[0,1] for j = 2, ..., d, one has

(21) |∂1,±f(x)−∂₁(f_m+f_m+Mm,3+n)(x)|< ε_m,n. Without limiting generality one can assume that %m,n ≤εm,n.

Set

R_n=

∞

[

m=1

Bk·k(f_m+f_m+Mm,3+n, %_m,n).

It is not difficult to see that R_n is open and dense in CC^d. Suppose that G∞ is the dense G_δ set from Proposition 9. Since G∞ is a subset of G from Proposition 8 all f ∈ G∞ are continuously differentiable on (0,1)^d. Moreover, the H¨older exponent is zero of these functions on the boundary of [0,1]^d.

Finally, set

G² =G∞

\

∞

\

n=1

Rn

! .

By construction, there exists a sequence of integers (m_n)n≥1 such that f ∈Bk·k(fmn+f_mn+Mmn,3+n, %n,mn) for every n.

For simplification, we setl_n=m_n+M_mn,3+n,ρ_n :=ρ_mn,n andε_n:=

ε_mn,n, so that for everyn≥1,ρ_n ≤ε_n= 2^−(ln)8,f ∈Bk·k(f_mn+f_ln, %_n), and for any x₁ ∈[ε_n,1−ε_n] and x_j ∈[0,1] for j = 2, ..., d

(22) |∂1,±f(x)−∂1(fmn +f_ln)(x)|< εn.

Proceeding towards a contradiction, suppose that there exists x = (x₁, x₂..., x_d) ∈ [0,1]^d where h_f(x) > 2. Since the H¨older exponent of f is zero on the boundary of [0,1]^d, necessarily x∈(0,1)^d.

Since hf(x)>2, one can find ε >0,D2, Cx ∈R such that for every small h,

(23) |f(x+he₁)−f(x)−∂₁f(x)h−D₂h²| ≤C_x|h|^2+ε.

Without limiting generality we can suppose thatε <1/2 holds as well.

Consider the unique integer j_n ∈ N such that x₁ ∈ [j_n2^−l2n,(j_n+ 1)2^−l2n). Next we consider two cases depending on whetherx1 ∈[jn2^−l2n,(jn+ 1)2^−l2n−2^−l4n], or x₁ ∈[(j_n+ 1)2^−ln2 −2^−l4n,(j_n+ 1)2^−l2n].

Case 1. Assume that x₁ ∈[j_n2^−ln2,(j_n+ 1)2^−l2n−2^−l4n] for infinitely many integers n≥1.

We set h_n = h_ne₁ with |h_n| = |h_n| = 2^−l2n/4, such that the first coordinates ofxandx+h_n both belong to [j_n2^−l2n,(j_n+ 1)2^−l2n−2^−l4n].

Combining (22), (23) and the fact that f ∈ Bk·k(f_mn +f_ln, %_n), we obtain

f_mn(x+h_n) +f_ln(x+h_n)

−

f_mn(x) +f_ln(x)

−h_n(∂₁f_mn+∂₁f_ln)(x)−D₂h²_n

≤ C_x|h_n|^2+ε+ 2%_n+ε_n|h_n|

≤ C_x|h_n|^2+ε+ 2·2^−(ln)8 + 2^−(ln)8|h_n|

≤ (Cx+ 1)|hn|^2+ε, (24)

where the last inequality holds since ρ_n ≤ ε_n ≤ 2^−(ln)8 << |h_n|³ for large n.

By using the Taylor polynomial estimate of the C^∞ function f_mn, one deduces that

(25)

f_mn(x+h_n)−f_mn(x)−∂₁f_mn(x)h_n−∂₁²f_mn(x) 2! h²_n

≤ M_mn,3 3! |h_n|³. Since by its definition ∂1f_ln(y) =γln(x1) (i.e. it is constant) for any y on the line segment connectingx and x+hn, we also have

(26) f_ln(x+h_n)−f_ln(x)−∂₁f_ln(x)h_n= 0.

Using (24), (25) and (26) we infer

∂₁²f_mn(x)

2! h²_n−D₂h²_n (27)

≤(C_x+ 1)|h_n|^2+ε+M_mn,3

3! |h_n|³ <(C_x+ 2)|h_n|^2+ε

where, using the fact that ε < 1/2, the last inequality holds if n is sufficiently large since M_n,3 ≤l_n ≤2^ln ≤ |h_n|^−1/2 forn large.

Now take h_n = 8|h_n|e₁. Then (24) and (25) used with h_n instead of hn for sufficiently largen yield

f_ln(x+h_n)−f_ln(x)−∂₁f_ln(x)|h_n|+∂₁²f_mn(x)

2! |h_n|²−D₂|h_n|²

≤ M_mn,3

3! |h_n|³+ (C_x+ 1)|h_n|^2+ε

=|h_n|^2+ε(M_mn,3

3! |h_n|^1−ε+C_x+ 1)

≤ |h_n|^2+ε(C_x+ 2).

(28)

Now for large n, it follows from (27) and |h_n|<|h_n|that

f_ln(x+h_n)−f_ln(x)−∂₁f_ln(x)|h_n|

≤ |h_n|^2+ε(2C_x+ 4).

(29)

Next we obtain a contradiction by using a lower estimate of the left- hand side of (29). By convexity of f_ln we have ∂₁f_ln(y) ≥ ∂₁f_ln(x) when y is on the line segment connecting x and x+h_n. Even more, this interval contains a subinterval of length larger than |h_n|/8 =|h_n| where

∂₁f_ln(y) = γ_ln(y) =∂₁f_ln(x) + 2^−l2n−ln =γ_ln(x₁) + 2^−l2n−ln. Thus,

f_ln(x+h_n)−f_ln(x)≥∂₁f_ln(x)|h_n|+|h_n|

8 ·2^−l2n−ln. By (29), one should have

|h_n|

8 ·2^−l2n−ln ≤ |h_n|^2+ε(2C_x+ 4).

Since 8|h_n|=|h_n|= 2·2^−l2n we would obtain that 2^−ln2−ln ≤(2·2^−l2n)^1+ε(2C_x+ 4), a contradiction when n is large.

Case 2. Suppose that x1 ∈ [(jn + 1)2^−l2n −2^−l4n,(jn+ 1)2^−l2n] for infinitely manyn ≥1.

We set h_n = h_ne₁ with |h_n| = |h_n| = 2^−l4n/2, such that the first coordinates of x and x+h_n both belong to [(j_n+ 1)2^−l2n−2^−ln4,(j_n+ 1)2^−l2n].

Since ρ_n and ε_n are still much smaller than |h_n|, equations (24) and (25) still hold, but now (26) is replaced by

(30) f_ln(x+h_n)−f_ln(x)−∂₁f_ln(x)h_n= ∂₁²f_ln(x)

2! h²_n= 2^{ln4−ln−l2n}h²_n 2 .

Using the same arguments as before but with (30), we deduce that

∂₁²f_mn(x)

2! h²_n−D₂h²_n+ 2^{l4n−ln−l2n}h²_n 2

<(C_x+ 2)|h_n|^2+ε.

This last inequality becomes impossible when n becomes large, since

|^∂12fmn₂ ^(x)| ≤Mmn,3 ≤ln≤2^l3n. Hence a contradiction.

4. The upper estimate

We start with the upper bound for the Hausdorff dimensions of the sets E_f^≤(h).

Proposition 11. If 1 < h ≤ 2 and f ∈ CC^d, then dimE_f^≤(h) ≤ d+h−2.

Proof. It is sufficient to treat the case h∈(1,2).

Assume that dimE_f^≤(h)> d+h−2.

For every x ∈E_f^≤(h), by Lemma 7, there exists a cone of direction C_ix, where i_x ∈ {1, ..., d} such that for everyz∈C_ix, the restriction of f to the straight line passing through xparallel tozhas exponent less than h.

Let us call E_i the set of elements of E_f^≤(h) satisfying this property with i_x =i ∈ {1, ..., N}. Obviously, E_f^≤(h) = SN

i=1E_i, so there exists at least one i∈ {1, ..., N} such that dimE_i > d+h−2.

Let us recall the following special case of Marstrand’s slicing theorem, Theorem 10.10 in Chapter 10 of [7]. Recall that S_d is the unit sphere inR^d.

Theorem 12. LetE ⊂[0,1]^d be a Borel set with Hausdorff dimension α ∈(d−1, d). Then for almost every z∈S_d (in the sense of (d−1)- dimensional “surface” measure), there exists a setE_zof positive(d−1)- dimensional Hausdorff measure in the hyperplane orthogonal to z such that for every x∈E_z, dimE∩(x+Rz) =α−(d−1).

EachC_ihas non-empty interior in the subspace topology ofS_d, hence it is of positive d−1-dimensional measure. Applying Theorem 12 to E_i, one can findz∈C_i and x∈[0,1]^d such that if D= (x+Rz), then dimE_i∩ D ≥d+h−2−(d−1) =h−1.

Let us call g the restriction of f to D. Then g is still a convex function of one variable.

By definition of Ei, every x∈ D ∩Ei satisfies hg(x)≤h.

Next, applying Lemma 6 tog, we deduce that min(h_g⁰

+(x), h_g⁰

−(x))≤ h−1, for every x∈ D ∩E_i.

Hence, at least one of the two sets E_g^≤0

+(h−1) and E_g^≤0

−(h−1) has Hausdorff dimension strictly greater than h−1.

But this is impossible, since both functionsg₊⁰ andg₋⁰ are monotone, and for such functions, by (4), the Hausdorff dimension of E_g^≤0

+(h− 1) and E_g^≤0

−(h− 1) is necessarily less than h− 1 ∈ [0,1]. Hence a contradiction, and the conclusion that dimE_f^≤(h)≤d+h−2.

Proposition 13. If 0≤h≤1, f ∈ CC^d, then dimE_f^≤(h)≤d−1.

Proof. The proof is immediate: iff ∈ CC^d, the pointwise exponent off at any x∈(0,1)^d is necessarily larger or equal than 1. The remaining points are located on the boundary, whose dimension isd−1. And for every h ∈ [0,1], it is easy to build examples of convex functions such that h_f(x) = hfor every xsatisfying x₁ = 0, so the upper boundd−1 for the Hausdorff dimension of E_f^≤(h) is optimal.

5. The lower estimate

Using Lemma 4 it is rather easy to “integrate” the result about functions in M¹ =M to obtain the one-dimensional result.

Theorem 14. There is a dense G_δ set G in CC¹ such that for any f ∈ G, and for any 1≤h≤2, dimE_f(h) = h−1.

Proof. Suppose 0< δ₀ <1/2 is fixed.

Recalling the result on typical monotone continuous functions, there exists aGδ set of functionsG^M,δ0 in the setM^1,δ0 :={f : [δ0,1−δ0]→ R, f monotone} such that every f ∈ G^M,δ0 satisfies (3).

Let us writeG^M,δ0 =T

n≥1G_n^M,δ0, where G_n^M,δ0 is a dense open set in M^1,δ0.

Let us choose a dense sequence (g_n,m)^∞_m=1 in G_n^M,δ0.

By taking antiderivatives of the elements of this sequence and by a suitable definition on the intervals [0, δ₀)∪(1−δ₀,1], one can choose a sequence of convex functions (f_n,m,k)^+∞_m,k=1 which is dense in CC¹ such that for every m, k ≥ 1, f_n,m,k⁰ (x) =g_n,m(x) for x ∈[δ₀,1−δ₀]. These functions f_n,m,k are continuously differentiable on [δ₀,1−δ₀].

Now, choose ε_n,m >0 such that

Bk·k(gn,m, εn,m)⊂G_n^M,δ0,

where the ball Bk·k is taken in the set M^1,δ0 using the L^∞-norm.

Lemma 4 gives the existence of %_n,m,k > 0 such that for every f ∈ Bk·k(f_n,m,k, %_n,m,k)⊂ CC¹, the inequality

|f_±⁰ (x)−g_n,m(x)|< ε_n,m

holds for all x∈[δ0,1−δ0].

Let us now introduce G_n=[

n,k

Bk·k(f_n,m,k, %_n,m,k).

By construction, G_n is dense in CC¹.

Proposition 8 yields the existence of a dense G_δ set D in CC¹ con- sisting of functions f continuously differentiable on (0,1).

We finally set

G =D\

∞

\

n=1

G_n

! . Clearly,G is a dense G_δ set in CC¹.

Suppose that f ∈ G, and set g =f⁰ on (0,1) andg₀ =g|_[δ0,1−δ₀]. Theng₀ ∈T∞

n=1G_n^M,δ0 and equation (3) implies that for any 1≤h≤ 2, dimE_g0(h−1) =h−1.

But Lemma 5 yields, E_g0(h−1) =E_f|_[δ

0,1−δ0](h), hence dimE_f(h) = h−1 for any 1≤h≤2.

Since δ₀ can be chosen arbitrarily small by taking a sequence of δ₀’s tending to zero, we can conclude the proof of the theorem.

Next we turn to the higher dimensional case.

Theorem 15. There is a denseG_δ setG inCC^dsuch that for anyf ∈ G and1≤h≤2, one has dimE_f(h)≥h+d−2. In addition,E_f(h) =∅ for h >2, E_f(h)∩[0,1]^d=∅ for 0< h < 1, and ∂([0,1]^d) =E_f(0).

Proof. Now instead of the functions, we can “integrate” the proof used in Section 4 of [4]. The idea is again to reduce the problem to the one-dimensional case. We select one coordinate direction, for ease of notation the first, the x₁-axis. We use in our proof the perturbation functions which are constant in the directions of the coordinate axes x_j,j = 2, ..., d already used in this paper, given in Definition 3.

Let us select a dense set of C^∞ functions {f_m : m = 1,2, ...} which is dense in CC^d. The x₁-partial derivatives, ∂₁f_m, of these functions will be denoted by g_m. The important feature of these functions g_m is the fact that they are monotone increasing in the x₁-variable. As our example at the beginning of the paper shows (see Remark 1), these functions are not necessarily monotone in the other variables, and this is why one cannot “integrate” simply the MISV genericity results.

Now, as in [4] and in the proof of Proposition 10, we use the functions γ_l and the perturbations f_l defined in (19) and (20).

Next, apply Lemma 3 to the functions f_m +f_m+n with ε = ε_n+m and j = 1. There exists a constant, denoted by %_m,n > 0, such that