1 Introduction
As nowadays is well known, the ramified structures which characterize river basins, blood vessels, leaves, trees and so on may be derived directly from a variational principle without relying on assumptions based on empirical observations (see [14, 22]). Indeed, if a cost functional is introduced which encourages joint transportation, the ramified structure comes out as the result of a compromise between the convenience of keeping together the fibers and the necessity of reaching a measure spread out on a large set. These recent developments, known as branched transport or irrigation models, can be included in the literature on the Monge transport problem, even if they are radically distinct from the approach proposed by Monge in [16]. Indeed, in the Monge–Kantorovitch model (see [10, 16]), the cost of the motion of a single particle is not influenced by interactions with the remaining part. In the context of irrigation models, one is not so much interested in knowing the final destination of a single particle (the so called “who goes where” problem) as in obtaining some information about the shape of the set of the trajectories, knowing, in particular, if particles move together giving rise to a river. The distinction is clearly reflected in the fact that V-shaped trajectories, optimal for the Monge–Kantorovitch functional, become Y-shaped ones, i.e. branching paths, for the irrigation functional. The branched transport has been introduced in [22, 14], where the existence of minima in an appropriate context is also proved. Further results are contained in [1, 12, 13, 3, 4, 20].
From the point of view of the regularity problem, one has two completely different issues: the regularity of the curves which describe the branches out of the branching points, and the properties of the branching structure (fractal regularity). An answer to the first question is given in [17], where the authors prove that the derivative of the particles trajectories for an optimal irrigation pattern have a locally bounded variation (if the irrigation measure is the Lebesgue measure restricted to a sufficiently regular set). The first result about fractal regularity is the multiscale result given in [6, Theorem 6.17], in which the authors prove that, under suitable assumptions and modulo constants, the number of branches of length ε starting from a branch having length ℓ is ℓε. An alternative way to approach the fractal regularity is the proof that some significant sets connected to branched networks in the Euclidean space ℝd have a fractal dimension.
This approach has led Pegon, Santambrogio and Xia to introduce in [18] the notion of unit (volume) irrigation ball and to investigate the dimension of its boundary (irrigation sphere in the present paper). They conjecture, see [18, Conjecture 4.3], that such a dimension should be d-β, where β=d(α-1)+1 is a relevant constant in the irrigation problem, and they partially prove the assertion showing in [18, Theorem 4.2] that, if 1-1d<α≤1, the upper Minkowski dimension of the boundary of the unit volume ball is less than or equal to d-β. Unfortunately, this inequality still leaves open the possibility that the sphere is a smooth surface of dimension d-1 and the conjecture is supported in [18] essentially by numerical computations.
The main aim of this paper is to supply more theoretical evidences by looking to some other significant sets which have a fractal dimension. We shall also propose a third approach to the fractal regularity by establishing oscillation estimates which are closely related to the scale transition approach in [6] (indeed the main result of this type is a consequence of [6, Theorem 6.17]) and to the approach in [18] because they are a fundamental tool for the dimension estimates. We shall recall the notion of what we call irrigation ball, we shall discuss some of its geometrical aspects and we shall introduce a natural notion of sub-balls (with the corresponding spheres). We shall prove some estimates on the lower and upper Minkowski dimensions of the spheres, which generically (both from a measure theoretic and from a topological point of view) hold true; see Theorem 2.5 and Corollary 2.7, respectively. Such results guarantee that the irrigation sub-spheres do not behave as regular sets, in the sense that, generically, the unique possible Minkowski dimension they may have is d-β. In addiction, we prove that the graph of the landscape function associated to an irrigation sphere (another significant set) has a Minkowski dimension equal to d+1-β.
The estimates on the landscape function oscillations have been a key tool to get the results and are, as already said, a further description of fractal regularity. We actually prove that the landscape function behaves as a Weierstrass-type function (see [21, 9]), well studied in the literature (see [2] and the references therein). In this regard it is worth remarking that the landscape function related to an optimal irrigation pattern, under the assumptions which will be introduced in Section 3, represents a significant example of a Weierstrass-type function which naturally arises in the study of branching structures and does not come from any artificial construction.
The paper is organized as follows: the main results will be formulated at the end of Section 2, after we recall the notion of irrigation ball introduced in [18], underline its geometrical aspects and propose a natural notion of concentric sub-ball of a given radius and related sphere. This preliminary part will give us the necessary notions needed for the statements. Section 3 is devoted to study the oscillations of the landscape function. In Section 4 we prove how the results obtained in the previous section allow to estimate the dimension of the level sets of a Hölder continuous function. Finally, in Section 5 we provide the proof of the main results.
2 Irrigation Balls and Concentric Sub-balls
The reader is supposed to be familiar with the literature on irrigation models or branched transport (see [1, 14, 12, 13, 3, 4, 20, 22]). Since the notation in the literature is not uniform, we shall list some of the symbols and basic notions we shall mainly use in the following. The references to the papers in which the reader can find a more detailed explanation are not necessarily the place in which each notion has been introduced for the first time:
χ denotes any irrigation pattern; see [6, Definition 1.1].
Branch of a pattern; see [7, Definition 6.2].
D
χ
is the flow zone and Dχ¯ is the domain of the pattern; see [6, Definition 1.8].
⪯
is the flow ordering; see [6, Definition 2.6].
ℓ
(
x
,
y
)
is the branch distance; see [6, Definition 2.9].
ℓ
(
x
)
is the residual distance; see [6, Definition 2.3].
I
α
(
χ
)
is the cost of the pattern; see [6, Definition 1.5] for i=0.
μ
χ
is the irrigation measure; see [7, Definition 2.15].
Optimal pattern is any irrigation pattern which minimizes Iα among the competitors with the same irrigation measure; see [7, Definition 4.1].
Simple pattern is any irrigation pattern with no cycle; see [7, Definition 6.1].
μ
(
x
)
is the mass flowing trough a point; see [6, Formula (1.1) (with i=0) and Remark 1.12].
Z is the (Santambrogio) landscape function introduced in [19]; see also [6, Definition 1.9].
For any Euclidean set E⊂ℝd, we shall denote by |E| its (outer) Lebesgue measure (in the following, the reader can safely assume to work always with measurable sets and functions if he finds it more relaxing). Moreover, we shall denote by dim¯M(E) and dim¯M(E) the lower and upper, respectively, Minkowski dimension of the set E (see [15, Section 5.3]) and, when the two dimensions agree, we shall denote by dimM(E) the Minkowski dimension of E. (The Minkowski dimension also has alternative names. It is sometimes called box-counting, metric, fractal or capacity dimension. Finally, the notion of Minkowski dimension has been extended to measures in [8, Definition 1.7]).
Given 1-1d<α≤1, we define as irrigation ball of unit volume and prescribed center x0∈ℝd, a solution of the minimization problem consisting in finding an irrigation pattern χ of minimum cost Iα(χ) among those which have source S=x0 and irrigation measure μχ equal to the restriction of the Lebesgue measure to a measurable set ℬ such that |ℬ|=1. The existence of such a χ is proved in [18, Theorem 2.1], the uniqueness is out of question if α<1 because χ cannot be invariant under rotations around x0. Sometimes we shall use the term irrigation ball in order to denote the set ℬ (which is also not unique) when the context makes the use not ambiguous.
When an irrigation unit volume ball χ is given, it defines a landscape function Z (see [19, 5, 18]). In [18, Theorem 2.3] it is proved that Z takes a constant value Z∗ on ∂ℬ, where
Z
∗
:=
e
α
α
(
α
+
1
d
)
,
with
e
α
:=
I
α
(
χ
)
.
We shall call Z∗ the radius of the unit volume irrigation ball and we shall consider Z as the radial distance function from the center x0. The scaling law in [18, Lemma 2.2] shows that for any given constant V>0 the problem of minimizing Iα with the constraint |ℬ|=V, i.e. of finding an irrigation ball of volume V, can be solved by scaling χ by a factor λ=V1/d, so obtaining a new pattern χV such that
(2.1)
I
α
(
χ
V
)
=
λ
α
d
+
1
e
α
=
V
α
+
1
d
e
α
and a new value of the corresponding landscape function on ∂ℬ given by
(2.2)
R
(
V
)
=
V
α
+
1
d
-
1
Z
∗
=
V
β
d
Z
∗
.
The last equality gives the radius of the irrigation ball in terms of the volume. Inverting such a function, we find
(2.3)
V
(
R
)
=
(
R
Z
∗
)
d
β
,
which gives the volume of an irrigation ball as a function of the radius R. In particular, we can compute the measure bα of the unit ball (ball of radius 1) as
(2.4)
b
α
=
(
1
Z
∗
)
d
β
=
(
α
e
α
(
α
+
1
d
)
)
d
β
=
(
α
d
e
α
(
α
d
+
1
)
)
d
β
,
which we prefer to use as a fundamental constant instead of eα for its more geometric meaning. So (2.2) and (2.3) respectively become
(2.5)
R
(
V
)
=
(
V
b
α
)
β
d
,
V
(
R
)
=
b
α
R
d
β
.
We finally observe that the least irrigation cost for an irrigation ball of radius R is given by eα(R)=Iα(χV(R)) and can be computed by (2.1), (2.4) and (2.5) as
e
α
(
R
)
=
(
V
(
R
)
)
α
+
1
d
e
α
=
(
b
α
R
d
β
)
α
+
1
d
e
α
=
α
b
α
β
d
(
α
+
1
d
)
(
b
α
R
d
β
)
α
+
1
d
=
α
α
+
1
d
b
α
α
+
1
-
β
d
R
α
d
+
1
β
=
α
d
α
d
+
1
b
α
R
d
β
+
1
=
α
d
α
d
+
1
R
V
(
R
)
,
which shows that the mean value of the landscape function on the ball ℬ is obtained by multiplying the radius R by αdαd+1. One could be tempted to define the irrigation ball by fixing a radius, intended as an upper bound on the landscape function, and maximizing the volume, but this choice would not lead to an optimal pattern. Indeed, adding a useless length to the fibers which end at a small landscape level would help to keep the maximum level low. So the definition in [18] looks to be the most reasonable one.
The above geometrical setting widely motivates the following definition.
Definition 2.1.
Let x0∈ℝd and R>0 and let ℬ be an irrigation ball centered in x0 and having radius R. For any 0≤ρ≤R the sets
ℬ
ρ
:=
{
x
∈
ℬ
∣
Z
(
x
)
<
ρ
}
,
ℬ
¯
ρ
:=
{
x
∈
ℬ
∣
Z
(
x
)
≤
ρ
}
,
𝒮
ρ
:=
{
x
∈
ℬ
∣
Z
(
x
)
=
ρ
}
will be respectively called (concentric) open, closed sub-ball and sub-sphere of ℬ of radius ρ.
It is worth explicitly remarking that, for α=1, the branched cost functional reduces to the usual Monge–Kantorovitch one, and so the notion of balls, concentric sub-balls and sub-spheres reduces to the usual one while the landscape function is nothing else than the Euclidean distance from the center (the source). Of course, in such a case, [18, Conjecture 4.3] is trivially true. On the contrary, when α<1, sub-balls are not irrigation balls of lower radius. We shall see soon that, for ρ=R, we have ℬρ=ℬ modulo a negligible set. We shall introduce functions defined on the interval [0,R] by setting for all ρ∈[0,R],
(2.6)
m
(
ρ
)
:=
|
ℬ
ρ
|
and (temporarily)
m
¯
(
ρ
)
:=
|
ℬ
¯
ρ
|
.
The functions m and m¯ are increasing, m is lower semicontinuous (left continuous) and m¯ is upper semicontinuous (right continuous). In Section 5 we shall prove the following statements.
Proposition 2.2.
For any ρ∈[0,R] we have |Sρ|=0.
Corollary 2.3.
For any ρ∈[0,R] we have m(ρ)=m¯(ρ) (so we will not use m¯ anymore) and m is a continuous function.
Proposition 2.4.
For any ρ∈[0,R] we have Sρ=∂Bρ.
We are now in a position to state the main results in the paper in which we shall implicitly assume that an irrigation ball ℬ of radius R is given and we shall use the notation introduced above.
Theorem 2.5.
For a.e. ρ∈]0,R],
dim
¯
M
(
𝒮
ρ
)
≤
d
-
β
.
Theorem 2.6.
There exists a dense Gδ-set G⊂[0,R] such that for all ρ∈G,
dim
¯
M
(
𝒮
ρ
)
≤
d
-
β
≤
dim
¯
M
(
𝒮
ρ
)
.
Corollary 2.7.
If ρ∈G as in Theorem 2.6, one of the two following alternatives holds:
dim
¯
M
(
𝒮
ρ
)
<
dim
¯
M
(
𝒮
ρ
)
(i.e. the irrigation sphere
𝒮
ρ
has not a Minkowski dimension).
dim
M
(
𝒮
ρ
)
=
d
-
β
.
Theorem 2.8.
The graph of the landscape function has a Minkowski dimension equal to d-β+1.
3 Lower Oscillation Estimates on the Landscape Function
Definition 3.1.
Let A⊂ℝd and let f:A→ℝ be a function. We call the quantity
ω
B
(
f
)
:=
sup
J
⊂
f
(
B
)
J
interval
|
J
|
the connected oscillation of f on B⊂A.
Note that ωB(f)≤oscBf:=supBf-infBf for any B⊂A and the equality holds if f is continuous and B is a connected set. Note that in this section and in the next one β will denote a generic exponent in [0,1] (and we do not specify this every time), while in Proposition 3.9 (and related formulas) and in the last section, in which we shall go back to the irrigation problem, β will be taken as 1+d(α-1) as specified in Section 1.
Definition 3.2.
Let A⊂ℝd be given. We say that a function f:A→ℝ is Hölder continuous with exponent β if there exists CH>0 such that
(HC)
|
f
(
x
)
-
f
(
y
)
|
≤
C
H
|
x
-
y
|
β
for every x,y∈A. We say that f satisfies the lower Hölder condition if there exists a constant CL>0 such that for every r>0 the connected oscillation of f on (the trace on A of) every ball Br of radius r centered in a point of A satisfies
(LHC)
ω
B
r
(
f
)
≥
C
L
r
β
.
Note that if A is a convex set and f is a continuous function, we get the usual definition of lower Hölder condition given in the literature on Weierstrass-type functions (defined on intervals). Note also that condition (LHC) with exponent β<1 implies nondifferentiability (i.e. nonexistence of a finite derivative) of the function at every point. We give a local version of the previous definitions.
Definition 3.3.
Let A⊂ℝd be given. We say that a function f:A→ℝ is Hölder continuous at a low scale with exponent β>0 if there exists a constant R¯>0 such that (HC) holds true for every x,y∈A such that |x-y|≤R¯. Analogously, we say that f satisfies the lower Hölder condition at a low scale with exponent β>0 if there exists a constant R¯>0 such that (LHC) holds true for (the trace on A of) every ball B with radius r≤R¯ centered in a point of A.
Sometimes we shall emphasize the constant R¯, which can of course be changed by varying the constants CH and CL, by writing that (HC) or (LHC) is satisfied, with a given constant, at the scale R¯.
Hölder continuous functions can easily be extended to larger sets. The result is probably better known for Lipschitz continuous functions, one just has to observe that Hölder continuity is the Lipschitz continuity with respect to the metric d(x,y)=|x-y|β. A slightly less simple variant, which we shall use to avoid a useless loss of generality, allows to extend a Hölder continuous function at a low scale defined on a set A to a neighborhood Ns(A) for a suitable s>0 (depending on the scale R¯), where, in correspondence to any s>0 and X⊂ℝd, we denote by
N
s
(
X
)
:=
{
x
∈
ℝ
d
∣
d
(
x
,
X
)
<
s
}
the s-neighborhood of the set X.
Let A⊂ℝd and let f:A→ℝ be a function which satisfies (HC) at a low scale R¯. For all x∈ℝd we set
(3.1)
φ
(
x
)
:=
inf
z
∈
A
|
x
-
z
|
<
2
3
R
¯
(
f
(
z
)
+
3
β
2
β
-
1
C
H
|
x
-
z
|
β
)
.
Lemma 3.4.
If z1,z2∈A and x∈Rd are such that
(3.2)
|
z
1
-
x
|
≤
1
2
|
z
2
-
x
|
<
1
3
R
¯
,
then
f
(
z
1
)
+
3
β
C
H
2
β
-
1
|
z
1
-
x
|
≤
f
(
z
2
)
+
3
β
C
H
2
β
-
1
|
z
2
-
x
|
.
Proof.
We can assume without any restriction that x=0, and so from (3.2) we deduce |z1-z2|≤32|z2|≤R¯. Moreover
|
z
2
|
β
-
|
z
1
|
β
≥
(
1
-
2
-
β
)
|
z
2
|
β
≥
(
1
-
2
-
β
)
(
2
3
|
z
1
-
z
2
|
)
β
=
2
β
-
1
3
β
|
z
1
-
z
2
|
β
.
So, by (HC), we get f(z1)-f(z2)≤CH|z1-z2|β≤3βCH2β-1(|z2|β-|z1|β). ∎
Corollary 3.5.
For any 0<s<R¯3, if x∈Ns(A), then
(3.3)
φ
(
x
)
=
inf
z
∈
A
|
x
-
z
|
<
2
s
(
f
(
z
)
+
3
β
C
H
2
β
-
1
|
x
-
z
|
β
)
.
Corollary 3.6.
The function φ is an extension of f, i.e. φ|A=f.
Proposition 3.7.
The restriction of the function φ to the set NR¯/6(A) is Hölder continuous at scale R¯3 with a constant 3βCH2β-1.
Proof.
Set s=R¯6 and fix x1,x2∈Ns(A) such that |x1-x2|≤R¯3. Then, for all z∈A such that |z-x1|<2s, we have |z-x2|≤2s+R¯3<23R¯, and so, by (3.1),
φ
(
x
2
)
-
(
f
(
z
)
+
3
β
C
H
2
β
-
1
|
x
1
-
z
|
β
)
≤
f
(
z
)
+
3
β
C
H
2
β
-
1
|
x
2
-
z
|
β
-
(
f
(
z
)
+
3
β
C
H
2
β
-
1
|
x
1
-
z
|
β
)
≤
3
β
C
H
2
β
-
1
|
x
1
-
x
2
|
β
.
Then, by taking the supremum with respect to z and by applying (3.3), we deduce
φ
(
x
2
)
-
φ
(
x
1
)
≤
3
β
C
H
2
β
-
1
|
x
1
-
x
2
|
β
.
∎
We also recall some definitions concerning the regularity of a measure μ.
Definition 3.8.
A measure μ is Ahlfors regular from below in dimension d if there exists a constant cA>0 such that
(LAR)
μ
(
B
(
x
,
r
)
)
≥
c
A
r
d
for all
r
∈
[
0
,
1
]
,
x
∈
supp
(
μ
)
,
while μ is Ahlfors regular from above in dimension d if there exists a constant CA>0 such that
(UAR)
μ
(
B
(
x
,
r
)
)
≤
C
A
r
d
for all
r
>
0
.
We shall say that μ is Ahlfors regular (in dimension d) if it is Ahlfors regular both from above and below. Finally, we shall say that μ is inner lower (resp. upper) Ahlfors regular on a set A⊂ℝd if (LAR) (resp. (UAR)) holds on balls contained in A.
In what follows we shall take a set ∅≠B⊂Dχ¯ which satisfies the following conditions:
(A1)
for all
x
,
y
∈
D
χ
such that
y
⪯
x
:
if
x
∈
B
, then
y
∈
B
,
which states that the set B is backward-stable with respect to the flow induced by the pattern χ,
(A2)
μ
χ
is (UAR) and inner (LAR) on
B
,
(A3)
for all
x
,
y
∈
D
χ
∩
B
such that
y
⪯
x
:
if
dist
(
x
,
∂
B
)
≤
ℓ
(
y
,
x
)
, then
Z
(
y
)
+
C
B
(
ℓ
(
y
,
x
)
)
β
≤
Z
(
x
)
,
(A4)
inf
∂
B
Z
≥
C
Z
(
μ
χ
(
ℝ
d
)
)
β
d
,
where CB and CZ are given positive constants and μχ(ℝd)=μ(x0) represents the total mass of the irrigation measure, i.e. the mass in the source. The main aim of this section is the proof of the following statement.
Proposition 3.9.
Let Z satisfy (HC) with β=1+d(α-1) on a set ∅≠B⊂Dχ¯ which satisfies conditions (A1), (A2), (A3) and (A4). Then Z satisfies (LHC) with the same value of β at a scale R¯ and with a constant CL which depends on α, the dimension d, the global mass μχ(Rd) and the constants CH, cA, CA, CB and CZ respectively appearing in (HC), (LAR), (UAR), (A3) and (A4).
In the following, we shall assume that such values are given, so that the expression universal constant will be intended as a constant which only depends on the above quantities.
We recall that the variation of the landscape function between two points x and y∈Dχ which are comparable by the flow order ⪯ is obtained by the following relation:
(3.4)
Z
(
x
)
-
Z
(
y
)
=
∫
y
x
(
μ
(
z
)
)
α
-
1
𝑑
ℋ
1
(
z
)
for all
y
⪯
x
,
where μ(z) denotes the mass flowing in the point z. Proposition 3.9 is a consequence of the fractal regularity result [6, Theorem 6.17], whose application is not straightforward just because we are only assuming the inner version of (LAR) instead of the global one. Note that the constant ε0 appearing in that statement is quantified in the proof and it can be written as
ε
0
:=
c
α
2
(
C
A
c
A
2
d
)
α
-
2
ℓ
=
C
T
ℓ
,
where cA and CA are as above and cα=α(1-α)2. So CT is a scale transition universal constant and ℓ represents the length of the main branch. The proof shows that the role of condition (LAR) consists in an estimate from below on the measure of a tubular neighborhood of a branch Γ. Of course the inner version of (LAR) works in the same way when we assume, as we shall do, that such a neighborhood is contained in B. In [6], (LAR) is assumed globally, but, of course, all the estimates which are quantified in terms of constants which do not involve cA but only depend on the other constants (in particular CH and CA) can be used without any restriction in this setting. Finally, we warn the reader that some estimates in [6] seem to explicitly depend on cA while they actually depend on CH. Indeed, the Hölder continuity of Z is usually derived from (LAR) (see [5, Theorem 6.2]), while we are assuming it directly in the statement of Proposition 3.9.
Proof of Proposition 3.9.
Set C1:=CT-1CA1/d>0, R¯=C1-1(μχ(ℝd))1/d and fix 0<r≤R¯ and x¯∈B. We shall initially assume that
(3.5)
μ
(
y
)
≤
(
C
1
r
)
d
for all
y
⪯
x
¯
such that
ℓ
(
y
,
x
¯
)
<
r
.
In particular, by our choice of R¯, we have ℓ(x0,x¯)>r. Then, starting from x¯, we can proceed backward along the flow, toward the source x0, and find y⪯x¯ such that ℓ(y,x¯)=r. Then, by applying (3.4) and (3.5), we get that Z(x¯)-Z(y)≥C1β-1rβ and we can conclude the proof easily. On the other hand, (3.5) does not hold when
(3.6)
there exists
y
¯
⪯
x
¯
and
ℓ
(
y
¯
,
x
¯
)
<
r
such that
μ
(
y
¯
)
>
(
C
1
r
)
d
.
Then set rT:=CT-1r. As a consequence, ℓ(y¯)>rT (indeed, otherwise condition (A2) would imply
μ
(
y
¯
)
≤
μ
χ
(
B
ℓ
(
y
¯
)
(
y
¯
)
)
≤
C
A
(
ℓ
(
y
¯
)
)
d
≤
C
A
(
r
T
)
d
=
(
C
1
r
)
d
,
in contradiction to (3.6)). Then, by applying [6, Lemma 2.17], we deduce the existence of a branch Γ¯ starting from y¯ having length rT. Let
N
r
(
Γ
¯
)
:=
{
z
∈
ℝ
d
∣
dist
(
z
,
Γ
¯
)
<
r
}
be the tubular neighborhood of Γ¯ of radius r. We shall assume that Nr(Γ¯)⊂B, in such a way to only use the inner version of (LAR) in order to estimate μχ(Nr(Γ¯)). Then we can argue as in [6, Theorem 6.17], getting the existence of a branch Γ~, starting from a point of Γ and with a length ℓ~ such that r≤ℓ~≤Wr, where W is the “scale window” universal constant introduced in [6]. So we can deduce, by applying as usual (UAR), that μ(z)≤CA(Wr)d for all z∈Γ~. Then, by taking two points x1⪯x2∈Γ~ such that ℓ(x1,x2)≥r, by applying (3.4), we obtain that ωΓ~(Z)≥CAα-1Wβ-1rβ. For the arbitrariness of r we can conclude that (LHC) holds.
If our extra assumption in this proof is false, we can deduce that the ball B0 centered in x¯ with radius (2+CT-1)r is not contained in B. So, for all ε>0 there exists a point xε∈Dχ∩B0 such that dist(xε,∂B)<ε. Then, starting from xε, if it is possible, we proceed backward along the flow of a distance r up to reach a point yε belonging to the ball B1 centered in x¯ with radius (3+CT-1)r. Since by construction ℓ(xε,yε)=r, by (A3) we get that ωB1(Z)≥Z(xε)-Z(yε)≥CBrβ, so (LHC) holds (under a suitable choice of CL). Finally, note that we cannot reach yε only when, during the backward motion, we find the source x0, and so, since dist(xε,∂B)<ε and Z(x0)=0, we get, by (HC) and (A4) and by our choice of R¯, that
ω
B
1
(
Z
)
≥
Z
(
x
ε
)
≥
inf
∂
B
Z
-
C
H
ε
β
≥
C
Z
C
1
β
R
¯
β
-
C
H
ε
β
.
So, for ε small, we see that (LHC) holds in every case (under a suitable choice of CL). ∎
Proposition 3.10.
Condition (A3) follows from the following variant:
(A${3^{\prime}}$)
there exists
C
B
′
>
0
such that for all
x
∈
B
∩
D
χ
we have
ℓ
(
x
)
≤
C
B
′
dist
(
x
,
∂
B
)
.
Proof.
Fix y⪯x∈B such that dist(x,∂B)≤ℓ(y,x)=:ℓ. Then for all y⪯z⪯x we have that dist(z,∂B)≤2ℓ, so by combining this last inequality with ((A${3^{\prime}}$)), we deduce ℓ(z)≤2CB′ℓ. Since μχ is (UAR), we deduce that
μ
(
z
)
≤
μ
χ
(
B
2
C
B
′
ℓ
(
z
)
)
≤
C
A
(
2
C
B
′
ℓ
)
d
.
Therefore, by (3.4), since d(α-1)=β-1, condition (A3) follows with CB=CAα-1(2CB′)β-1. ∎
We conclude this section by giving an estimate on the margin of growth of the landscape function with respect to the residual distance.
Lemma 3.11.
If Z satisfies (HC) and μχ satisfies (UAR) globally, then, for a suitable constant CM>0,
for all
x
∈
D
χ
there exists
z
⪰
x
such that
Z
(
z
)
-
Z
(
x
)
≥
C
M
ℓ
(
x
)
β
.
Proof.
Fix x∈Dχ. Then, by applying [6, Lemma 2.17] (which, as already pointed out, can be applied thanks to the Hölder continuity of Z), for every ε>0 there exists yε⪰x such that ℓ(x,yε)=ℓ(x)-ε. By (3.4) and (UAR) we get
Z
(
y
ε
)
-
Z
(
x
)
=
∫
x
y
ε
(
μ
(
z
)
)
α
-
1
𝑑
ℋ
1
(
z
)
≥
C
A
α
-
1
ℓ
(
x
)
β
-
1
ℓ
(
x
,
y
ε
)
.
Then the thesis follows by taking CM<CAα-1 and ε small enough. ∎
4 Dimension Estimates on Level Sets
The main results of this paper will follow from two estimates respectively derived from properties (HC) and (LHC) at a low scale.
Fixing a function f:A→ℝ, for any a≤b∈ℝ we set
L
a
b
:=
{
x
∈
A
∣
a
≤
f
(
x
)
≤
b
}
and Lc=Lcc when a=b=c∈ℝ to denote a level set of the function f.
Lemma 4.1.
Let A⊂Rd and let f:A→R be a function which satisfies (HC) at scale R¯ with exponent β. Then, for any a<b∈R and for any
0
<
δ
<
C
H
2
β
-
1
(
2
β
-
1
)
R
¯
β
,
setting
h
=
(
(
2
β
-
1
)
δ
3
β
2
C
H
)
1
β
,
we have that, for any large k∈N, if (ci)1≤i≤k⊂[a+δ2,b-δ2] is any family of levels such that
(4.1)
|
c
i
-
c
j
|
≥
δ
for all
i
≠
j
,
then
(4.2)
∑
i
=
1
k
|
N
h
(
L
c
i
)
|
≤
|
N
h
(
L
a
b
)
|
.
Proof.
Set Ni:=Nh(Lci) for all i. We shall prove the thesis by showing that
(4.3)
N
i
∩
A
⊂
L
a
b
for all
i
∈
{
1
,
…
,
k
}
,
and
(4.4)
N
i
∩
N
j
=
∅
for all
i
,
j
∈
{
1
,
…
,
k
}
,
i
≠
j
.
By fixing i and x∈Ni∩A, there exists some x¯i∈Lci such that |x-x¯i|<h<R¯. Then, by (HC), we deduce that
|
f
(
x
)
-
c
i
|
≤
C
H
|
x
-
x
¯
i
|
β
<
δ
2
(
2
β
-
1
3
β
)
<
δ
2
.
So, since ci∈[a+δ2,b-δ2], we deduce that x∈Lab and, by the arbitrariness of x, we deduce (4.3). As proved in Proposition 3.7, the restriction of f to the set Lab can be extended to Nh(Lab) if h is small enough (i.e. k is large enough) in such a way to remain Hölder continuous at scale R¯3 with a constant 3β2β-1CH. So, by identifying f with its extension, we can prove (4.4) by showing that f(Ni)∩f(Nj)=∅ for i≠j. If, by contradiction, f(Ni)∩f(Nj)≠∅ for some i≠j, we get the existence of xi∈Ni and xj∈Nj such that f(xi)=f(xj) and the existence of x¯i∈Lci and x¯j∈Lcj such that
max
(
|
x
i
-
x
¯
i
|
,
|
x
j
-
x
¯
j
|
)
<
h
<
R
¯
6
.
Then, by Hölder continuity, max(|f(xi)-ci|,|f(xj)-cj|)<CHhβ≤δ2. Then, since f(xi)=f(xj), by the triangular inequality, we get a contradiction to (4.1). ∎
A corresponding statement which gives the opposite estimate is the following lemma.
Lemma 4.2.
Let A⊂Rd and let f:A→R be a function which satisfies (LHC) at scale R¯ with exponent β. Then, for any a<b∈R and for any 0<δ<CLR¯β, setting h=(δ2CL)1/β, we have that for any k∈N, if (ci)1≤i≤k is a δ2-net of [a,b], then
(4.5)
∑
i
=
1
k
|
N
h
(
L
c
i
)
|
≥
|
L
a
b
|
.
Proof.
Set Ni:=Nh(Lci) for all i. The thesis follows since, by construction, (Ni)1≤i≤k is a covering of Lab. Indeed, fixing x∈Lab, since by assumption h<R¯, and setting B=B(x,h), by (LHC) we have that ωB(f)≥CLhβ=δ2, and so f(B(x,h)∩A) must contain an interval whose length is at least δ2. Then, since (ci)1≤i≤k is a δ2-net of [a,b], there exists y∈B such that f(y)=ci for some i∈{1,…,k}. Then, since |x-y|<h, we deduce that x∈Ni. ∎
The two following propositions are easy consequences of the above lemmas.
Proposition 4.3.
Let A⊂Rd and let f:A→R be a function which satisfies (HC) at scale R¯ with exponent β. Let h¯>0, γ<β and a<b∈R such that |Nh¯(Lab)|<+∞. Then there exist an open set V⊂[a,b] and a positive constant h<h¯ such that
(4.6)
|
N
h
(
L
c
)
|
h
γ
≤
1
for all
c
∈
V
.
Proof.
Fix k∈ℕ, k≥1 and set δ=b-a2k. We can split the interval [a,b] into 2k+1 sub-intervals Vi⊂[a,b] such that the first and last interval V1 and V2k+1 have length |V1|=|V2k+1|=δ2 and the remaining 2k-1 have length δ. Setting
h
=
(
(
2
β
-
1
)
δ
3
β
2
C
H
)
1
β
,
we claim that for every k large enough there exists ı¯∈{1,…,k} such that
(4.7)
|
N
h
(
L
c
)
|
≤
|
N
h
(
L
a
b
)
|
k
for all
c
∈
V
2
ı
¯
.
Indeed, on the contrary, for all i∈{1,…,k} would exist ci∈V2i such that |Nh(Lci)|>|Nh(Lab)|/k, i.e. there would exist a finite family of levels (ci)1≤i≤k which, by construction, is contained in [a+δ2,b-δ2] and satisfies (4.1). Therefore, ∑i=1k|Nh(Lci)|>|Nh(Lab)|, but since, for k large enough,
δ
=
b
-
a
2
k
<
C
H
2
β
-
1
(
2
β
-
1
)
R
¯
β
and Lemma 4.1 applies, we get a contradiction to (4.2).
So we take V=V2ı¯ as in (4.7). For all c∈V,
|
N
h
(
L
c
)
|
≤
|
N
h
(
L
a
b
)
|
k
≤
4
C
H
b
-
a
3
β
2
β
-
1
h
β
|
N
h
(
L
a
b
)
|
=
4
C
H
b
-
a
3
β
2
β
-
1
h
β
-
γ
h
γ
|
N
h
(
L
a
b
)
|
,
and so (4.6) follows since the constant 4CHb-a3β2β-1hβ-γ|Nh(Lab)| is less than 1 (since β>γ and h is small) when k is large enough. ∎
Proposition 4.4.
Let A⊂Rd and let f:A→R be a function which satisfies (LHC) at scale R¯ with exponent β∈]0,1]. Let h¯>0, γ<β and a<b∈R such that |Lab|>0. Then there exist an open set V⊂[a,b] and a positive constant h<h¯ such that
(4.8)
|
N
h
(
L
c
)
|
h
γ
≥
1
for all
c
∈
V
.
Proof.
Fix k∈ℕ, k≥1 and set δ=3(b-a)k. We can split the interval [a,b] into 2k+1 sub-intervals Vi⊂[a,b] such that the first and last interval V1 and V2k+1 have length |V1|=|V2k+1|=δ12 and the remaining 2k-1 have length δ6. Setting
h
=
(
δ
2
C
L
)
1
β
,
we claim that for every k large enough there exists ı¯∈{1,…,k} such that
(4.9)
|
N
h
(
L
c
)
|
≥
|
L
a
b
|
k
for all
c
∈
V
2
ı
¯
.
Indeed, on the contrary, for all i∈{1,…,k} would exist ci∈V2i such that |Nh(Lci)|≤|Lab|/k, i.e. there would exist a finite family of levels (ci)1≤i≤k which, by construction, is a δ2-net of [a,b] and satisfies ∑i=1k|Nh(Lci)|≤|Lab|. Since, for k large enough,
δ
=
3
(
b
-
a
)
k
<
C
L
R
¯
β
,
and Lemma 4.2 applies, we get a contradiction to (4.5).
So we take V=V2ı¯ as in (4.9). For all c∈V,
|
N
h
(
L
c
)
|
≥
2
C
L
3
(
b
-
a
)
h
β
|
L
a
b
|
=
2
C
L
3
(
b
-
a
)
h
β
-
γ
h
γ
|
L
a
b
|
,
and so (4.8) follows since the constant 2CL3(b-a)hβ-γ|Lab| is greater than 1 (since β<γ and h is small) when k is large enough. ∎
Given a measurable set A⊂ℝd and a real-valued function f:A→ℝ, we define F:A×ℝ→ℝ by setting F(x,y):=f(x)-y for all x∈A and y∈ℝ. Of course we have that the graph of f is the zero level set of the function F. We point out that if f satisfies (LHC) (at a low scale), so does F with the same constant CL and the same scale R¯. On the other hand, if f is (HC), then F is only (HC) (with the same exponent β) at a low scale for any R¯>0 with a different constant C~H. Finally, note that, if f is measurable, the level sets of the function F can easily be estimated as
(4.10)
|
L
a
b
|
=
(
b
-
a
)
|
A
|
for all
a
<
b
∈
ℝ
.
For a general f just one of the two inequalities holds and only the lower bound to |Lab| given by (4.10) can be applied.
Theorem 4.5.
Let A⊂Rd be a measurable set such that |Ns(A)|<+∞ for some s>0. Let f:A→R be Hölder continuous with exponent β. Then
(4.11)
dim
¯
M
(
graph
f
)
≤
d
+
1
-
β
.
Proof.
Fix k∈ℕ suitably large and let {-1<c1<…<c2k+1<1} be a partition of the interval [-1,1] such that c1+1=1-c2k+1=12k+1 and ci+1-ci=22k+1 for all i∈{1,…,2k}. Then fix a real number R¯>0 such that the function F is (HC) at scale R¯ for some constant C~H. We can apply Lemma 4.1 to the function F (with d replaced by d+1) by taking a=-1, b=1 and k large enough to get that δ:=12k+1<C~HR¯β. Then set
h
=
(
δ
2
C
~
H
)
1
β
.
By a translation in the y-variable of F we easily see that the value of |Nh(Lci)| is the same for all indexes i, so (4.2) and (4.10) give |Nh(Lci)|≤12k+1|Nh(L-11)|=22k+1|Nh(A)| for all i∈{1,…,2k+1}. Since, by construction ck+1=0, we get that
|
N
h
(
graph
f
)
|
≤
1
2
k
+
1
|
N
h
(
L
-
1
1
)
|
=
2
2
k
+
1
|
N
h
(
A
)
|
≤
4
C
~
H
|
N
h
(
A
)
|
h
β
since
h
=
(
δ
2
C
~
H
)
1
β
=
(
1
2
(
2
k
+
1
)
C
~
H
)
1
β
,
and so 12k+1=2C~Hhβ. The values assumed by h for k varying in ℕ are dense enough to estimate the limit for h→0. So we get an upper bound on the Minkowski content (see [15, Section 5.5])
M
c
s
(
graph
f
)
:=
lim
h
→
0
+
|
N
h
(
graph
f
)
|
h
d
+
1
-
s
≤
2
C
~
H
|
N
h
(
L
-
1
1
)
|
=
4
C
~
H
|
N
h
(
A
)
|
when the dimension s equals d+1-β. ∎
Theorem 4.6.
Let A⊂Rd be a measurable set such that |A|>0. Let f:A→R satisfy the lower Hölder condition with exponent β. Then
(4.12)
dim
¯
M
(
graph
f
)
≥
d
+
1
-
β
.
Proof.
The proof follows the same argument already used to prove Theorem 4.5 by applying Lemma 4.2 instead of Lemma 4.1 and using (4.10) as a lower bound. ∎
By combining (4.12) with (4.11), we get the following result.
Corollary 4.7.
The graph of any function defined on a measurable set A⊂Rd such that |A|>0 and |Ns(A)|<+∞ for some s>0, which satisfies (HC) and (LHC) with the same exponent β, has a Minkowski dimension equal to d+1-β.
We introduce some notation. Fixing h¯>0 and γ∈ℝ, we set
S
-
(
γ
,
h
¯
)
:=
{
c
∈
f
(
A
)
∣
there exists
h
<
h
¯
such that
|
N
h
(
L
c
)
|
≤
h
γ
}
,
S
+
(
γ
,
h
¯
)
:=
{
c
∈
f
(
A
)
∣
there exists
h
<
h
¯
such that
|
N
h
(
L
c
)
|
≥
h
γ
}
,
and state the following conditions on the measure of the sets |Lab| related to f:
(M${{}^{-}}$)
for all
a
,
b
∈
f
(
A
)
,
a
<
b
, there exists
s
>
0
such that
|
N
s
(
L
a
b
)
|
<
+
∞
,
(M${{}^{+}}$)
for all
a
,
b
∈
f
(
A
)
,
a
<
b
, we have
|
L
a
b
|
>
0
.
Remark 4.8.
Note that conditions ((M${{}^{-}}$)) and ((M${{}^{+}}$)) are trivially satisfied when A is bounded or, respectively, A is an open connected set and f is a continuous function.
Propositions 4.3 and 4.4 respectively guarantee that, if condition (M±) is satisfied and ±γ>±β, the inner part of the set S±(γ,h¯) is dense in f(A). So, setting S±(γ):=⋂h¯>0S±(γ,h¯), we get, by the Baire Theorem, that both S±(γ) are dense Gδ-sets if conditions (M±) are both satisfied. Obviously, by construction, if c∈S-(γ), then dim¯M(Lc)≤d-γ, while if c∈S+(γ), then dim¯M(Lc)≥d-γ.
Finally, the sets
𝒮
±
:=
⋂
±
γ
>
±
β
γ
∈
ℚ
S
±
(
γ
)
are dense Gδ-sets if conditions (M±) are true.
Theorem 4.9.
Let A⊂Rd and let f:A→R be a real-valued function defined on A which satisfies (HC) with exponent β. Then, if condition ((M${{}^{-}}$)) holds true, there exists a dense Gδ-set S⊂f(A) such that
(4.13)
dim
¯
M
(
L
c
)
≤
d
-
β
for all
c
∈
𝒮
.
Proof.
Just take as 𝒮 in (4.13) the set 𝒮-. ∎
Theorem 4.10.
Let A⊂Rd and let f:A→R be a real-valued function defined on A which satisfies (LHC) (at a low scale) with exponent β. Then, if condition ((M${{}^{+}}$)) holds true, there exists a dense Gδ-set S⊂f(A) such that
(4.14)
dim
¯
M
(
L
c
)
≥
d
-
β
for all
c
∈
𝒮
.
Proof.
Just take as 𝒮 in (4.14) the set 𝒮+. ∎
Finally, by combining Theorem 4.9 with Theorem 4.10, we get the following corollary which states a topological generic estimate on the Minkowski dimension (when it exists) of the level sets of a function which satisfies (HC) and (LHC) with the same exponent β.
Corollary 4.11.
Let A⊂Rd be an open set and let f:A→R be a real-valued function defined on A which satisfies (HC) and (LHC) (at a low scale) with exponent β. Then, if both (M±) are true, there exists a dense Gδ-set S⊂f(A) such that
dim
¯
M
(
L
c
)
≤
d
-
β
≤
dim
¯
M
(
L
c
)
for all
c
∈
𝒮
.
As a consequence, if c∈S and there exists dimM(Lc), then dimM(Lc)=d-β.
Proof.
Just take 𝒮=𝒮-∩𝒮+, which is still a dense Gδ-set, and combine (4.13) with (4.14). ∎
5 Conclusions
To the aim to apply to the irrigation ball ℬ and to its landscape function Z the results in the previous section, we need to preliminarily establish property (LHC), and therefore to prove the assumptions needed in Section 3.
By [18, Theorem 3.7], the landscape function Z satisfies (HC) with exponent β=1+d(α-1). Moreover, since μχ is the restriction of the Lebesgue measure to the irrigation ball ℬ, we have that Z satisfies (UAR) globally and the inner (LAR) on ℬ (with the same constants CA and cA). So, ℬ satisfies (A2) and, trivially, (A1). Condition (A4) follows from (2.5) with CZ=bα-β/d.
Proof.
Fix x∈ℬ. Since Z=R on ∂ℬ, we deduce by (HC) that R-Z(x)≤CHdist(x,∂B)β. On the other hand, by Lemma 3.11, there exists z⪰x such that Z(x)≤Z(z)-CMℓ(x)β≤R-CMℓ(x)β. By combining the two inequalities above, we get CMℓ(x)β≤CHdist(x,∂ℬ)β, and so the thesis follows with CB′=(CHCM-1)1/β. ∎
By Proposition 3.10, Z also satisfies (A3). So, by Proposition 3.9, Z satisfies (LHC) on ℬ. Since the irrigation sub-spheres are level sets of Z, the main results of this paper (stated in Section 2) follow from Section 4.
Proof of Proposition 2.2.
We prove that any 𝒮ρ does not contain Lebesgue points. Let x¯∈𝒮ρ and B=B(x¯,r) with r<R¯. By Proposition 3.9 we deduce the existence of a point x∈B such that |Z(x)-Z(x¯)|≥CLrβ. Then, setting r′:=(CL2CH)1/βr (which is of the same order as r), we have B(x,r′)∩𝒮ρ=∅. ∎
Proof of Proposition 2.4.
The inclusion ∂ℬρ⊂𝒮ρ is trivial. On the other hand, fixing x¯∈𝒮ρ, we shall prove that for all (small) δ>0 such that |x¯-x0|>δ in the Euclidean ball Bδ(x¯) there exists a point y such that Z(y)<ρ, i.e. y∈ℬρ. Indeed, fixing 0<ε<δ, we get a point x∈Dχ∩Bε(x¯), and since x0∉Bδ(x), we can fix a point y which precedes x in the flow order such that y∈∂Bδ(x¯). Obviously, we have that ℓ(y,x)≥δ-ε, and so we get, by (3.4), Z(x)-Z(y)≥|ℬ|α-1ℓ(y,x)≥|ℬ|α-1(ε-δ). Therefore, since Z satisfies (HC),
Z
(
y
)
≤
Z
(
y
)
-
Z
(
x
)
+
osc
B
(
x
0
,
ε
)
Z
+
|
Z
(
x
0
)
|
≤
ρ
+
C
H
ε
β
-
|
ℬ
|
α
-
1
(
δ
-
ε
)
.
So, by letting ε go to zero, we have Z(y)<ρ. ∎
Proof of Theorem 2.5.
Given ρ∈[0,R] and δ>0, setting h=(δCH)1/β, we deduce that
|
N
h
(
L
c
)
|
≤
|
L
c
-
δ
c
+
δ
|
.
On the other hand, by Definition 2.6, we have that |Lc-δc+δ|=m(c+δ)-m(c-δ). Then, since 1hβ=CHδ, we deduce that
(5.1)
|
N
h
(
L
c
)
|
h
β
≤
C
H
[
m
(
c
+
δ
)
-
m
(
c
)
δ
+
m
(
c
)
-
m
(
c
-
δ
)
δ
]
.
Since the function m is monotone, by the Lebesgue theorem on differentiability of monotone functions (see [11]), for a.e. c the following upper Dini derivative is finite:
D
¯
m
(
c
)
:=
lim sup
σ
→
0
+
m
(
c
+
σ
)
-
m
(
c
)
σ
.
So, we deduce by (5.1) that for a.e. c,
lim sup
h
→
0
|
N
h
(
L
c
)
|
h
β
≤
2
C
H
D
¯
m
(
c
)
<
+
∞
,
and therefore that dim¯M(𝒮ρ)=dim¯M(Lc)≤d-β. ∎
Finally, since Z is continuous on ℬ, which is a bounded open connected set, both conditions (M±) are satisfied (see Remark 4.8). Then Theorem 2.6 and Theorem 2.8 respectively follow from Corollary 4.11 and Corollary 4.7 applied to the landscape function Z.
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and
Sergio Solimini
