On the dimension of an irrigable measure

On the Dimension of an Irrigable Measure.

G. DEVILLANOVA(*) - S. SOLIMINI(**)

ABSTRACT- In this paper the problem of determining if a given measure is irrigable, in the sense of [4], or not is addressed. A notion of irrigability dimension of a measure is given and lower and upper bounds are proved in terms of the minimal Hausdorff and respectively Minkowski dimension of a set on which the measure is concentrated.

A notion of resolution dimension of a measure based on its discrete approx- imationsis alsointroduced andits relation withthe irrigation dimension is studied.

Introduction.

In the paper [4] the authors have introduced a cost functional to the aim of modeling ramified structures, suchas trees, root systems, lungs and cardiovascular systems. A very similar functional (even if the variable employed has a different form) has been introduced in [10]. The aim of the functional is to force the fibers to keep themselves together penalizing, in this way, their branching. The necessity of keeping the functional low competes with a boundary condition which, on the other hand, forces the fibers to bifurcate prescribing that the fluid they carry must reach a given measure spread out on a volume. The result of this competition is that the fibers take advantage in keeping themselves together as long as possible and then branching, always into a finite number of branches, while ap- proaching the terminal points, giving rise to the ramified structure. In [10]

the problem consisting in determining the cases, depending on an index, in which all the probability measures can be reached by a system of fibers (an irrigation pattern) of finite cost, i.e. are irrigable measures, is formulated and solved in a very close setting.

(*) Indirizzo dell'A.: Laboratoire du CMLA, Ecole Normale SupeÂrieure de Cachan, France.

(**) Indirizzo dell'A.: Dipartimento di Matematica, Politecnico di Bari, Via Orabona, 4 - 70125 Bari, Italy.

In this work we shall investigate a more general question consisting in characterizing, for a given value of the index, what probability measures are irrigable or not. The answer to this question will clearly show, in particular, what are the cases in which all the probability measures will turn out to be irrigable, giving in this way a different proof of the already mentioned result in [10]. The fact that a measure spread on a set of high dimension forces the fibers to a more frequent branching, and therefore needs a higher cost, seems to suggest that the higher it is the dimension on which a measure is spread the more difficult it becomes to irrigate it. For a better formalization, we introduce the notion of irrigability dimension of a measure and then we equivalently express the above stated problem in terms of giving some estimates on the irrigability dimension of a given positive measure which is always supposed to be Borel regular, with a bounded support and a finite mass (by normalization we shall suppose it to be a probability measure). We shall show, with some examples, that the intuitive and conjecturable idea that the irrigability dimension of a mea- sure coincides withthe Hausdorff dimension of its support is groundless in spite of the fact that both the two values express how much the measure is spread out. On the other hand, we shall give some lower and upper bounds for the irrigability dimensiond(m) of a probability measuremby means of the minimal Hausdorff and respectively Minkowski dimension of a set on which the measure is concentrated.

This result will be overproved. Indeed, we shall prove it directly, getting some further meaningful information and introducing some tools which will be also used in other parts of the paper but we shall also be able to deduce it from a deeper estimate of d(m) which will need the in- troduction of new notions. More precisely, it will need the notion of re- solution dimensionof a measure which, affected by an index, expresses the possibility to describe the measure by means of discrete approxima- tions. When the measure is suitably regular, the value of the resolution dimension does not depend on the index, while for a generic measure, as will be explained by some examples, the resolution dimension is ``out of focus'' in the sense that different indexes give different values. We shall show that, in any case, it is always possible to find an index, suitably characterized, which gives a resolution dimension which coincides with the irrigability dimension.

The paper is organized as follows: In Section 1 we shall introduce the notion of irrigability dimension and we shall state the main results which do not make use of the notion of resolution dimension of a measure.

Sections 2 and 3 are respectively dedicated to the lower and upper esti-

mate given ford(m) by means of the minimum among the Hausdorff and the Minkowski dimension of the sets on which the measure is con- centrated. In Section 4 remarks and examples, mainly based on the compactness results stated in [4], which show that the estimates are, in a certain sense, sharp are collected. In Section 5 we shall introduce the notion of resolution dimension of a measure and we shall state some fundamental properties. The proof of the irrigability and nonirrigability results which can be deduced from conditions on the resolution dimension will be respectively shown in sections 6 and 7. In Section 8 we shall show how the irrigability dimension of a measure can be seen as a resolution dimension withrespect to some index p1 and how to chose such a suitable value ofp. Then we shall give another proof of the main result in Section A (Theorem 1.1).

Since we are dealing withnotions introduced for the first time in [4]

and [3], which will be used without any explanation, in order to help the reader we have gathered up in Appendix A the notation and the results in [4] and [3] which are essential for the understanding of this paper. In Appendix B we give the proof of the propositions stated in Section 5 with some examples which justify the required assumptions. Finally, in Ap- pendix C we give the index of the main notation.

1. Dimensions of a measure and irrigability results.

We just recall the definition of irrigation pattern while, as said in the introduction, we have gathered up in Appendix A the notation and the results in [4] and [3] which will help the reader for the understanding of this paper.

Let (V; j j) be a nonatomic probability space which we interpret as the reference configuration of a fluid material body. We can also interpret it as the trunk section of a tree, this trunk being thought of as a set of fibers which can bifurcate into branches. A set ofVwith source point S2R^Nis a mapping

x:VR_‡!R^N suchthat:

C1) For a.e. material point p2V, x_p(t):t7!x(p;t) is a Lipschitz continuous map with a Lipschitz constant less than or equal to one.

C2) For a.e.p2V:x_p(0)ˆS.

The conditionjVj ˆ1 is of course assumed by normalization in order to simplify the exposition. In some cases this normalization will be impossible (we can, for instance, work withtwo different spaces and assume an in- clusion), then we shall consider all the notions trivially extended to the case jVj<‡ 1. We shall consider the source point S2R^N as given and we shall denote byCS(V) andPS(V) the set of all the set of fibers of V and respectively the set of all the measurable set of fibers ofVand we shall call the elements ofP_S(V)irrigation patterns.

We shall introduce some definitions which will be used to formalize the irrigability problem.

DEFINITION1.1. For a fixed real numbera2]0;1[we shall call critical dimension of the exponentathe constant daˆ 1

1 a>1, i.e. the conjugate index 1

a

0

of the index1 a>1.

DEFINITION1.2. Leta2]0;1[be given and letmbe a probability mea- sure onR^N. We shall say thatmis an irrigable measure with respect toa(or that m is a-irrigable) if there exists a pattern x2P_S(V) of finite cost I_a(x)<‡ 1such thatm_xˆm.

It is clear that two approaches are possible and equivalent: one can fix a constanta2]0;1[ and investigate the irrigable measures with respect to this constant or fix a measuremand find out the constantsa2]0;1[ with respect to which m is irrigable. This second point of view leads us to in- troduce the following definition.

DEFINITION1.3. Letmbe a positive Borel measure onR^N, then we shall call irrigability dimension of mthe number

d(m)ˆinffdajm is irrigable withrespect toag:

REMARK1.1. For any probability measurem, by definition, the irrig- ability dimension d(m)ofmis greater or equal to1.

REMARK1.2. If m is an irrigable measure with respect to a, then m is also irrigable with respect to every constant b2[a;1[. Indeed, let x2PS(V) such that Ia(x)<‡ 1 and m_xˆm, then for all ba, Ib(x)Ia(x).

REMARK1.3. By the definition of d(m)and byRemark 1.2it follows that for a givena2]0;1[and for a given measurem:

1) if d(m)<dathenmisa-irrigable;

2) if d(m)>dathenmis nota-irrigable.

As we shall show in Section 4, both cases can occur whend_a ˆd(m), see examples 4.4 and 4.5.

The aim of the first part of this paper is to give operative estimates of d(m) in terms of geometrical properties of the measurem. So we introduce the following two definitions.

DEFINITION1.4. We shall say that a positive Borel measuremonR^Nis concentrated on a Borel set B ifm(R^NnB)ˆ0and we shall call \cd ofmthe smallest Hausdorff dimension d(B)of a set B on whichmis concentrated i.e. the number

d_c(m)ˆinffd(B)jmis concentrated on Bg:

DEFINITION1.5. We shall denote bysupp (m)the support ofmin the sense of distributions and shall callsupport dimension ofm, d_s(m), its Hausdorff dimension.

REMARK1.4. The support of a measure can be characterized as the smallest closed set on whichmis concentrated and the existence of such a set a priori follows by the separability of R^N, precisely by the Lin- deloÈf property. While, as stated above, the existence of the smallest closed set on which m is concentrated is granted, it is clear that the smallest set on whichmis concentrated, in general, does not exist. This is the reason for which the infimum is taken in Definition 1.4, even if a set B of minimal dimension on whichm is concentrated can always be fixed. Moreover being supp (m)a set on which mis concentrated, it fol- lows that

dc(m)ds(m):

These two geometrical dimensions are not sufficient to study the ir- rigability of a measure, as we shall show later in examples 4.1 and 4.3.

DEFINITION1.6. Let XR^Nbe a bounded set. We shall call Minkowski dimension of the set X (see[8]) the constant

d_M(X)ˆN lim inf

d!0 log_djN_d(X)j …1:1†

where, for alld>0,

N_d(X)ˆ fy2R^Njd(y;X)<dg: It is useful to remark that

0dM(X)N 8X6ˆ ;: …1:2†

Moreover the Minkowski dimension of a set XR^N can be char- acterized by the following two properties:

8b<dM(X) lim sup

d!0 jNd(X)jd^{b N}ˆ ‡ 1 …1:3†

and

8b>d_M(X) lim

d!0jN_d(X)jd^{b N}ˆ0: …1:4†

LEMMA1.1. Let X R^Nandb>d_M(X). Then we can cover X by using d ^b balls of radiusdfor alldsufficiently small.

PROOF. Beingb>dM(X), we have, by (1.4), that for allC>0 and for d>0 sufficiently small

jN^d

2(X)j Cd^{N b}:

We consider any family of disjoint balls (B_i)_i2I of radius d

2 contained in N^d

2(X). We know that, being, for alli2I,jB_ij ˆb_N d 2

N(b_Nstands for the

measure of the unitary ball ofR^N), card (I)bN d 2

N jN^d

2(X)j Cd^{N b} so, taking asCthe constantbN

2^N,

card (I)C2^N

b_Nd ^b ˆd ^b: …1:5†

We have shown that the number of elements of any family consisting of disjoint balls contained inNd

2(X) is bounded byd ^b. This allows us to find a family of such balls which is maximal by inclusion. The corresponding fa- mily of balls withthe same centers but withdouble radius, by maximality, turns out to be a covering ofX. Inequality (1.5) gives the thesis. p LEMMA 1.2. Let XR^N and b<dM(X). It is not possible to find a constant C>0such that one can cover X with only Cd ^bballs of radiusd for alldsufficiently small.

PROOF. We shall proceed by contradiction assuming that there exists a constant C>0 suchthat ford sufficiently small it is possible to cover X usingCd ^bballs of radiusd. It is useful to remark that doubling the radius of these balls we get a covering ofN_d(X), so we have

jNd(X)j Cd ^bbN(2d)^Ncostd^{N b};

which givesbd_M(X) by (1.3). p

REMARK1.5. Collecting the last two lemmas, we can say that for a set XR^N

…1:6† dM(X)ˆinffb0jX can be covered by Cbd ^b balls of

radiusdfor all d1g: DEFINITION1.7. Letmbe a probability measure, we shall use the no- tation

d_M(m)ˆinffd_M(X)jm is concentrated on Xg …1:7†

and we shall call itMinkowski dimension ofm.

REMARK1.6. For any subset X ofR^Nthe Minkowski dimensions of X and of its closure X are the same. Therefore

d_M(m)ˆd_M( supp (m)):

Moreover the Hausdorff dimension d(X) of a set X is less or equal to dM(X). So for any probability measurem

d_s(m)d_M(m): …1:8†

REMARK 1.7. Let m be a probability measure, then collecting Re- mark1.4and(1.8)we have that the following inequalities hold for ds(m)

d_c(m)d_s(m)d_M(m): …1:9†

A similar estimate is enjoyed by d(m). Indeed, we shall prove the fol- lowing statement.

THEOREM 1.1. (Lower and Upper bound on d(m)). Let m be a prob- ability measure then the following bounds hold for d(m)

d_c(m)d(m)maxfd_M(m);1g: …1:10†

The first inequality in (1.10) is a straightforward consequence of a

deeper and more precise result stated in the following theorem, whose proof is in Section 2.

THEOREM 1.2. Let a2]0;1[ and let m be an a-irrigable probability measure, thenmis concentrated on a d_a-negligible set, in particular,

d_c(m)d_a: …1:11†

Theorems 1.1 and 1.2 widely answer the question considered in [10]

about the values of awhich make every measure of bounded support ir- rigable. Indeed, we can deduce the following corollaries.

COROLLARY1.1. Leta2]0;1[,a> 1

N⁰. Then any probability measure mwith a bounded support isa-irrigable.

PROOF. Remarking thata> 1

N⁰ is equivalent todaˆ 1 a

0>N, com- bining (1.2) with(1.10), we have, for everym,

d_a>Nmaxfd_M(m);1g d(m);

so every probability measurem witha bounded support isa-irrigable by

Remark 1.3,1. p

COROLLARY1.2. Leta2]0;1[be such that any probability measurem with a bounded support isa-irrigable, thena> 1

N⁰.

PROOF. From Theorem 1.2 we have that any probability measure m witha bounded support is concentrated on ad_a-negligible set. So,N<d_a, namelya> 1

N⁰. p

In spite of inequalities (1.10) and (1.9) it is not possible to establishsome general inequality between d(m) and ds(m), as shown in Section 4 by ex- amples 4.1 and 4.3.

By the following lemmas we shall make the estimates on the dimension d(m) more precise in the case in which the probability measure m enjoys some regularity properties.

DEFINITION1.8. Letmbe a probability measure andb0. We shall say thatmis Ahlfors regular in dimensionbif

…AR† 9C₁;C₂>0s:t:8r2 ‰0;1Š; 8x2supp…m†:C₁r^bm…B…x;r†† C₂r^b:

We shall separately consider the two bounds in (AR). So for a prob- ability measuremand a real numberb0 we shall consider the two con- ditions

9C>0 s:t:8r2 ‰0;1Š; 8x2supp…m†: Cr^bm…B…x;r††: …LAR†

and

9C>0 s:t:8r2 ‰0;1Š; 8x2supp…m†: m…B…x;r†† Cr^b: …UAR†

In (UAR) the restrictionx2supp (m) can be removed, this could make the value ofC₂increase at most of 2^b. It is useful to recall the following defi- nition.

DEFINITION 1.9. A probability measure n:R^N!R_‡ satisfies the uniform density property (in short u.d.p.) in dimensionb0on a set M if

9C1>0 s.t.8x2M;8r2[0;1]: C1r^bn(B(x;r)):

LEMMA1.3. Letnbe a probability measure which satisfies the u.d.p. in dimensionb0on a subset B. Then

d_M(B)b: …1:12†

PROOF. Let us fixd>0 and let us consider any family (B_i)_i2Iof disjoint balls of radius d

2 withcenters on B. By hypotheses, n(B_i)C2 ^bd^b and n(B)1, therefore card (I)2^bC ¹d ^b. So we can consider a family (B_i)_i2I as above maximal by inclusion. The maximality of (B_i)_i2Iguarantees that, for any other pointx2B,d x; S

i2IB_i

<d

2holds. Therefore the family (B~_i)_i2I which is obtained by doubling the radius of the ballsBiis a covering ofB. So we have proved thatBcan be covered by constd ^bballs of radiusdarbi- trarily small and so by Remark 1.5d_M(B)b. p

COROLLARY1.3. Letmbe a probability measure. Letb0such thatm satisfies (LAR) (i.e.msatisfies the uniform density property in dimension b onsupp (m)). Then

d_M(m)b:

…1:13†

REMARK 1.8. The thesis of Corollary 1.3 still holds true if one as- sumes the existence of a probability measure n which satisfies the uniform density property in dimension b on a set B on which m is concentrated.

LEMMA 1.4. Let m be a probability measure concentrated on a set AR^N. Letb0such thatmsatisfies (UAR). Then

H^b(A)>0:

…1:14†

PROOF. Let (X_i)_i2I be any countable covering ofA. EveryX_iis con- tained in a ballB_iwitha radius equal to (X_i). So, by (UAR)

1ˆm(R^N)ˆm(A)X

i2I

m(B_i)CX

i2I

(X_i)^b ; from which we have

X

i2I

(X_i)^b C ¹>0:

p COROLLARY1.4. Letmbe a probability measure. Letb0such thatm satisfies (UAR). Then

d_c(m)b: …1:15†

COROLLARY1.5. Let m be an Ahlfors regular probability measure in dimension b1. By Corollary 1.3 and Corollary 1.4, being bˆmaxfb;1g maxfd_M(m);1g, the lower and upper bounds stated in Remark 1.7 and Theorem 1.1 for ds(m) and d(m) respectively, give

dc(m)ˆds(m)ˆd(m)ˆdM(m)ˆb:

This guarantees that, in the case of an Ahlfors regular probability measure, all the geometrical dimensions d_c(m), d_s(m)and d_M(m)and the irrigability dimension d(m)are equal to the Ahlfors dimensionb.

COROLLARY1.6. An Ahlfors regular probability measuremof dimen- sionb1, isa-irrigable for alla2]0;1[s.t.da>bi.e. for alla2]1

b⁰; 1[

and is not irrigable for alla2]0;1[s.t.dabi.e. for alla2]0; 1 b⁰].

PROOF. Let a2]0;1[. If da 6ˆbˆd(m), the thesis follows from Re- mark 1.3. Moreover, whendaˆb, by Theorem 1.2 it is clear that an Ahlfors regular probability measure of dimensionbˆd_a, is nota-irrigable. Indeed by Lemma 1.4 it cannot be concentrated on ad_a-negligible set. p We shall make use of this last argument when in Section 4 we shall show that, in general, d(m)ˆinffdajmisa-irrigableg is not a minimum, see Example 4.4.

2. Lower bound ond(m).

This section is devoted to the proof of Theorem 1.2 from which d_c(m)d(m) trivially follows.

LEMMA2.1. Letx2PS(V)be an irrigation pattern ofVand r>0, then m_x(R^NnB_r(S)) I_a(x)

r ¹_a

: …2:1†

PROOF. Taking into account that the less expensive way to carry some part of the fluid out of B_r(S) is to move it in a unique tube in the radial direction and to leave the other part in the source point, we have

[m_x(R^NnB_r(S))]^arI_a(x);

from which the thesis follows. p

COROLLARY 2.1. Let x2PS(V) be an irrigation pattern of V. If r(I_a(x))^{1 a}, then

m_x(R^NnB_r(S))I_a(x): …2:2†

In [3] the following lemma has been proved.

LEMMA2.2. Letx2PS(V)be a simple irrigation pattern ofVwithout dispersion and e>0, then there exists a finite number k2Nof points x_i2Fxsuch that, denoting byx_ithe branch ofxwith source point x_i,

X^k

iˆ1

I_a(x_i)<e …2:3†

m_x X^k

iˆ1

m_xi

(R^N)<e: …2:4†

LEMMA2.3. Letx2P_S(V)be a simple irrigation pattern ofVwithout dispersion ande>0, then9AR^Nsuch that

1) A can be covered by a finite number of balls B_iˆB_ri(x_i), s.t.

P

i (ri)^da<e;

2) m_x(R^NnA)e.

PROOF. By Lemma 2.2 we can find a finite number k2N of points xi2R^Nsuchthat, by denoting byx_ithe branch ofxstarting fromxiand by e_iˆIa(x_i), we have

X^k

iˆ1

e_i<e: …2:5†

Calling, for all i2 f1;. . .;kg, as suggested by Corollary 2.1, r_iˆ(I_a(x_i))^{1 a}ˆ(e_i)^{1 a}we have,

X

i

r^d_i^aˆX

i

e_i<e: Moreover, from (2:2), we can deduce that

m_xi(R^NnB_ri(x_i))e_i: Applying (2.4), (2.3) and (2.5) we get, forAˆ S^k

iˆ1B_ri(x_i), that m_x(R^NnA)(m_x X^k

iˆ1

m_xi)(R^N)‡X^k

iˆ1

m_xi(R^NnB_ri(x_i))<e‡X^k

iˆ1

e_i<2e: Replacingebye

2we complete the proof. p

PROOF OFTHEOREM1.2. By hypotheses there exists an irrigation pat- ternx2P_S(V) of finite costI_a(x)<‡ 1, suchthatmˆm_x. By Lemma A.2 we know thatd(F_x)ˆ1<d_a, therefore we can reduce ourselves, as Re- mark A.3 suggests, to a pattern x without dispersion. Moreover, if one considers a pattern which is optimal with respect to the cost functional, the pattern can also be supposed to be simple (see Definition A.7), see [3, Theorem 6.1].

So, for everyn2N, we can apply Lemma 2.3 to the patternxand to eˆ2 ⁿ>0. Therefore for alln2Nthere existsA_nR^Nwhich satisfies 1) and 2) of Lemma 2.3 foreˆ2 ⁿ. For a fixedh2Nwe shall denote by D_hˆ T

n>hAn

Then

m_x(R^NnD_h)ˆm_x [

n>h

R^NnA_n X

n>h

m_x(R^NnA_n)X

n>h

1 2ⁿˆ 1

2^h : …2:6†

Moreover, beingDhAnfor alln>h, by Lemma 2.3,1),Dhis covered by a finite number kof balls of radiusr_iverifying P^k

iˆ1r^d_i^a <2 ⁿ, from which

H^da(Dh)ˆ0 follows by the definition of Hausdorff outer measure. For all i2N

m_x

R^Nn [

h2N

D_h

m_x(R^NnD_i) 1 2ⁱ ; therefore we have

m_x

R^Nn [

h2N

D_h

ˆ0 and somis concentrated on S

h2ND_h. Since, for allh2N,H^da(D_h)ˆ0 we get thatmis concentrated on ada-negligible set. p PROOF OFTHEOREM1.1 (lower boundd_c(m)d(m)). By Theorem 1.2 we have proved in particular that, for everya2]0;1[, ifmisa-irrigable then dc(m)da. By the definition ofd(m), taking the infimum ondain the above

inequality, the thesis follows. p

3. Upper bound ond(m).

The main goal of this section is the proof of the following theorem, from which the upper bound ond(m) stated in Theorem 1.1 easily follows.

THEOREM3.1. Letmbe a probability measure anda2]0;1[, thenmis a-irrigable provided d_M(m)<d_a.

To this aim, we need to introduce some definitions and to establish some preliminary lemmas.

DEFINITION3.1. Let Iˆ f1;2;. . .;ng Nbe a finite set of indexes. We shall say that(P_i;g_i)_i2Iis a hierarchy of collectors if

8i2I : Piis a finite subset ofR^Nwith kielements xⁱ_j,1jki; 8i2I;i6ˆn; g_i maps P_i in P_i‡1 while g_n is a map on Pn of constant value S (which is the ``head'' of the hierarchy and will be the source S in the applications).

In the following we shall call each mapg_ithe ``dependence'' map of the pointsxⁱ_j2Pifrom thosex^i‡1_j 2Pi‡1.

REMARK3.1. For a given hierarchy of collectors(Pi;g_i)_i2I, every time we fix a point xˆx¹_j 2P1, we find, using the dependence maps, a chain of pointsfx;g₁(x);g₂(g₁(x));. . .;Sgwhich allows us to reach the source S ``in a hierarchical way''. We can consider the elements of such a chain as the vertices of a polygonal which runs with unitary speed. Reversing the time, we get a path which starts from the source S and arrives in x. We shall call by gx:R‡!R^N this path, parameterized in the whole of R‡, by con- sidering it constant after reaching x.

In what follows, let us set,8x2P₁,g¹(x)ˆg₁(x),g²(x)ˆg₂(g¹(x)) and recoursively

g^j(x)ˆg_j(g^{j 1}(x))2P_j‡1:

For a given hierarchy of collectors (Pi;g_i)_i2I, we shall deal with a probability measure m₁ concentrated on P1, namely m₁ˆP^k1

iˆ1m¹_jd_x¹

j is the sum of a finite number of Dirac masses centered on the pointsx¹_j ofP₁.

BeingVa non atomic probability space, by Lyapunov Theorem, we can splitVintok1(ˆcard(P1)) setsV_jsuchthatjV_jj ˆm¹_j, i.e. we can splitV intok1 sets whose measures are just the massesm¹_j we find in the points (x¹_j)_j2k1 at the base of the hierarchy.

DEFINITION 3.2. Let (Pi;g_i)_i2I be a hierarchy of collectors and m₁ˆP^k1

jˆ1m¹_jd_x¹

j a probability measure concentrated on the base P1 of the hierarchy.

We shall say thatx:VR‡!R^Nis a distribution pattern relative to m₁ and to the hierarchy(P_i;g_i)_i2Iif8p2V, and8t0:

x(p;t)ˆg_x¹

j(t) for p2V_j; where the paths g_x¹

j and the partition(V_j)_1jk1are as above.

REMARK3.2. Let(P_i;g_i)_i2Ibe a hierarchy of collectors andm₁ˆP^k1

jˆ1m¹_jd_x1 j

a probability measure concentrated on the base P₁of the hierarchy. Letx be a distribution pattern relative to m₁ and to the hierarchy (P_i;g_i)_i2I. Then, by construction, being ix(Vj)ˆ fx¹_jg, we have

m_x ˆX^k1

jˆ1

m¹_jd_x¹

j ˆm₁:

DEFINITION 3.3. Let (Pi;g_i)_1in be a hierarchy of collectors. For any discrete probability measurem₁concentrated on P1we shall recoursively call for all i2 f2;. . .;ng,m_ithe image measure ofm_{i 1}through the functiong_{i 1}. Eachone of these measures can be considered as a discrete measure defined on the whole of the space and concentrated onP_i.

LEMMA3.1. Let(Pi;g_i)_i2Ibe a hierarchy of collectors andm₁be a dis- crete probability measure concentrated on the base level P1 of the hier- archy. Letxbe a distribution pattern relative tom₁and to the hierarchy, then

Ia(x)ˆX

i2I

X

x2Pi

(m_i(x))^ajx g_i(x)j; …3:1†

where, for alli2I, and for allx2Pi

m_i(x)ˆm_i(fxg):

PROOF. We shall proceed by induction onnˆcard (I). The thesis is obvious in the case nˆ1. Let us suppose that the statement is true for card (I)ˆn 1 and let us prove that the statement is also true for card (I)ˆn. Let us remark that each one of thek_nˆcard (P_n) elements of the last level setPncan be seen as the head of a hierarchy ofn 1 levels, given by the setsPi(x), where for all 1in 1:

P_i(x)ˆ fy2P_ijg_{n 1}(g_{n 2}((g_i(y))))ˆxg: and by the suitable restrictions of the mapsg_i,iˆ1;. . .;n 1.

Therefore we can apply the induction hypotheses to thek_nbranchesx_x ofxwhich start from the pointx2Pn. So each one of these patterns has a cost which can be estimated by

Ia(x_x)ˆX^{n 1}

iˆ1

X

y2Pi(x)

(m_i(y))^ajx g_i(y)j:

To bringx_xback to the sourceSobtaining the patternx_x, restriction of xto S

x¹_j2P1(x)

V_j, we must add toI_a(x_x) the cost necessary for the connection ofxto the sourceS. Therefore we have

I_a(x_x)ˆ(m_n(x))^ajx Sj ‡I_a(x_x)

ˆ(m_n(x))^ajx g_n(x)j ‡X^{n 1}

iˆ1

X

y2Pi(x)

(m_i(y))^ajx g_i(y)j:

Since the wholexcan be regarded as a multiple branchstarting from the sourceSwhich has the patternsx_xas the corresponding single branches, by additivity we have

I_a(x)ˆX

x2Pn

I_a(x_x)ˆX

x2Pn

(m_n(x))^ajx g_n(x)j‡X

x2Pn

X^{n 1}

iˆ1

X

y2Pi(x)

(m_i(y))^ajx g_i(y)j

ˆXⁿ

iˆ1

X

x2Pi

(m_i(x))^ajx g_i(x)j:

p Lemma 3.1 admits the following corollary.

COROLLARY3.1. Let(P_i;g_i)_i2I be a hierarchy of collectors andm₁ be a probability measure concentrated on the base levelP1of the hierarchy. Let xbe a distribution pattern relative tom₁ and to the hierarchy, then

I_a(x)X

i2I

k¹_i ^al_i; where for all i2 f1;. . .;ng

liˆmax

x2Pijx g_i(x)j: PROOF. The thesis follows because for alli2I:

X

x2Pi

(m_i(x))^a(k_i)^{1 a}: Indeed, by HoÈlder inequality, being, for alli2I, P

x2Pi

mi(x)ˆ1, we have:

X

x2Pi

(m_i(x))^a X

x2Pi

m_i(x)

!_a X

x2Pi

1

!_{1 a}

ˆk¹_i ^a:

p PROOF OFTHEOREM3.1. Ifd_M(m)<d_a, we can fix a constantbsuchthat dM(m)<b<da.

Given n2N,n1, let us consider a covering of supp (m) consisting of balls withradius 2 ⁿ. Let us call X_n the set made of the centers of such balls and let us set X₀ˆ fSg. We introduce for n1 th e map W_n:X_n!X_{n 1}which chooses, for every pointx2X_n, one of th e closest pointsW_n(x)2Xn 1. It is easy to see that forn2 (and forn1, witha suitable choice ofSand a normalization of the diameter of the support

ofm)

8x2Xn : jx W_n(x)j 32 ⁿ: …3:2†

Moreover, by Lemma 1.1, being d_M(m)<b, we can choose X_n and a constant C>0 so th at

card (Xn)C(2 ⁿ) ^bˆC2^nb: …3:3†

Let us now put a total order on X_n. On eachcenterx2X_n we shall put the mass

mⁿ_xˆm(B2 ⁿ(x)n[

y<xB2ⁿ(y)): In this way we get a probability measurem_nˆ P

x2Xn

mⁿ_xd_xsuchthatm_n*m.

Now, for a fixedn2N, all 1in, let us callP_iˆX_{n i‡1}andg_iˆW_{n i‡1}. By (3.3) we have:

8i2 f1;. . .;ng:k_iˆcard (P_i)ˆcard (X_{n i‡1})C(2 ^{(n i‡1)}) ^b; …3:4†

while, by (3.2),

…3:5† 8i2f1;. . .;ng:l_iˆmax

x2Pijx g_i(x)jˆ max

x2Xn i‡1jx W_{n i‡1}(x)j3(2 ^{(n i‡1)}):

If we denote byx_n a distribution pattern relative to the hierarchy of collectors (P_i;g_i)_1inand tom₁ˆm_n, by Corollary 3.1, using also (3.4) and (3.5), we have

Ia(x_n)Xⁿ

iˆ1

(k_i)^{1 a}l_iC^{1 a} Xⁿ

iˆ1

[(2 ^{(n i‡1)}) ^b]^{1 a}3(2 ^{(n i‡1)})

ˆ3C^{1 a} Xⁿ

iˆ1

2 ^{(n i‡1)( b(1 a)‡1)}ˆ3C^{1 a} Xⁿ

jˆ1

2 ^jb3C^{1 a} 2^b 1 ; where, beingb<d_a, is

bˆ b(1 a)‡1>0:

The independence on n of the above bound allows us to build a se- quence of patterns (x_n)_n2N to which we can apply the compactness theo- rem [4, Theorem 8.1] and to get, in this way, the existence of a limit

patternxof finite cost suchthatm_xˆm. p

It is worth remarking that the measurem_n taken in the proof of The- orem 3.1 could be replaced by any probability measure centered on the

points ofXn such that the Kantorovitch-Wasserstein distance betweenm_n andm(see Definition 5.3) is less or equal to 2 ⁿ.

PROOF OF THEOREM1.1 (upper bound d(m)maxfd_M(m);1g). Ar- guing by contradiction, let us supposed(m)>maxfd_M(m);1g. Then there exists a constant a2]0;1[ suchthat dM(m)<da<d(m). From one side dM(m)<da, so we have from Theorem 3.1 that m is a-irrigable; on the other side da<d(m), so we get from Remark 1.3 (2) that mcannot bea-

irrigable. p

4. Remarks and examples.

DEFINITION4.1. Leta2]0;1[and letmbe a finite measure onR^N. We shall calla-cost of the measuremthe value of the functional Iaon the op- timal patternsxwhich irrigate the measurem.

LEMMA 4.1. Leta2]0;1[,nandmbe two finite measure onR^N such thatnm. Ifmisa-irrigable then alsonisa-irrigable, moreover thea-cost to irrigatenis less expensive than thea-cost form.

PROOF. For any n2N, let us consider a countable borel partition A_nˆ(Aⁿ_i)_i2IofR^Nmade of sets of diameter less or equal to¹_nfor alli2I.

By hypotheses there exists an irrigation pattern x, defined on VR‡, whereVis a probability space, s.t.Ia(x)<‡ 1andm_xˆm. Let us call, for alli,V_i;nˆi_x¹(Aⁿ_i). By construction we get

jV_i;nj ˆ ji_x¹(Aⁿ_i)j ˆm_x(Aⁿ_i)ˆm(Aⁿ_i)n(Aⁿ_i):

Therefore, being anyV_i;n a non atomic set, by Lyapunov Theorem, V_i;n admits a subsetV⁰_i;nsuchthatjV⁰_i;nj ˆn(Aⁿ_i). Let us considerV⁰_nˆS

i V⁰_i;n and let us denotex_nˆx_jV⁰

n. By constructionm_xn *nand

I_a(x_n)I_a(x)<‡ 1: …4:1†

Therefore, by compactness, we get a limit patternx suchthatm_xˆn.

Moreover, being Ia a lower semicontinuous functional, (4.1) gives Ia(x)lim inf

n!‡ 1Ia(x_n)Ia(x). p

The following corollaries easily follow.

COROLLARY 4.1. Let a2]0;1[,c2Rand n andm two finite Radon measures onR^N such thatncm. Then

d(n)d(m):

COROLLARY 4.2. Let m and n be finite Radon measures such that c₁mnc₂mfor some positive constantsc₁; c₂. Then we have

d(n)ˆd(m):

REMARK 4.1. The patternx, found in the proof of Lemma4.1, is the limit pattern, modulo equivalence, of a sequence of subpatterns ofxbut it is not a subpattern in general. So one could wonder if it is always possible to find a subpattern ofxwhich irrigatesn.

The answer to this question is negative. For instance, one can consider V ˆ[0;1]and, for a.e. p2[0;1]and for all t0,

x(p;t)ˆmin (p;t): …4:2†

It is clear thatxirrigates the Lebesgue measurem_Lon[0;1]. On the other side, it is not possible to find a subpattern ofxwhich can irrigatenˆ1

2m_L. Indeed, in such a case, one should find a subset A[0;1] of density 1 2 everywhere and this is not possible. A pattern x: 0;1

2

R_‡!R, pro- vided by the proof ofLemma 4.1is, for instance,

x(p;t)ˆmin (2p;t); …4:3†

which irrigates1 2m_L.

The simple idea that the irrigability of a probability measuremdepends only on the dimension of the support is false. Indeed,d_s(m) andd(m) are not comparable in general, even if the dimensionsd_c(m) andd_M(m) which re- spectively give a lower and an upper bound on ds(m) are also bounds for d(m), as stated in Remark 1.7 and Theorem 1.1.

It is easy to see that, in general,d_s(m)6d(m), as stated in the following example.

EXAMPLE4.1. There exist probability measuresmsuch thatd_s(m)ˆN (maximum possible value) and d(m)ˆ1 (minimum possible value) i.e.

which area-irrigable for alla2]0;1[.

PROOF. Let us call B the unit ball of R^N, Sˆ0 and let B~ ˆ

ˆ fx₁;x₂;. . .;x_n;. . .gbe the countable set consisting in the points ofBwith rational coordinates.

Let us consider mˆ P

n1

1 2

nd_n where, 8n2N, d_n is a Dirac mass centered inx_n. By construction,d_s(m)ˆN. Moreover we shall prove thatm isa- irrigable for alla2]0;1[, i.e.d(m)ˆ1. Letxbe the pattern which at unitary speed carries fromSin then-thpoint ofB~ the mass 1

2ⁿ. Then for anya2]0;1[ we have:I_a(x) P

n1

1 2

anˆ 1

2^a 1<‡ 1. Therefore, being

by constructionm_xˆm,misa-irrigable. p

In order to show that also the converse inequality is, in general, not true, we shall point out the following property.

PROPOSITION4.1. Leta2]0;1[andmbe a probability measure which is not a-irrigable. Then for all n2N it is possible to find a discrete approximationm~ofm, of sufficiently high resolution(see Definition 5.1), such that any pattern~xwhich irrigates~mhas a cost I_a(~x)n.

PROOF. Assume by contradiction that we can find a sequence (~m_n)_n2N of discrete approximations of m weakly converging to m and a sequence (~x_n)_n2N of patterns, where, 8n2N, ~x_n irrigates m~_n, suchthat, 8n2N:

Ia(~x_n)<c. Then we could apply the compactness theorem [4, Theorem 8.1]

obtaining a limit patternxof finite cost which irrigatesm. p EXAMPLE 4.2. There exists a probability m with a countable support which is nota-irrigable foraˆ 1

N⁰.

PROOF. LetBbe the unit ball ofR^N, letm_Lbe the Lebesgue measure on B and aˆ 1

N⁰. Being, by Theorem 1.2, m_L not a-irrigable, by Proposi- tion 4.1, we can consider a discretizationm₁ofm_Lsuchthat for any pattern

~x₁which irrigatesm₁:Ia(x₁)1. Analogously, letm₂be a discretization of 1

2m_Ldistributed on1

2B(ball of radius1

2) suchthat for any patternx₂which irrigatesm₂:I_a(x₂)2. Recoursively, for anyn2Nletm_nbe a discretiza- tion of 1

2ⁿm_Lrestricted to1

nBsuchthat for any patternx_nwhich irrigates it, I_a(x_n)n

…4:4†

holds true. Let mˆP

n1m_n (normalized, if we really want to produce a probability measure) and let us remark that supp (m)ˆ S

n1supp (m_n)[ f0g and therefore, being for alln1 supp (m_n) a finite set, supp (m) is countable.

Let us show that m is not a-irrigable. Indeed, the a-irrigability of m would imply, by Lemma 4.1 (beingm_nmfor alln1), that anym_n isa- irrigable witha bounded cost and this is in contradiction with(4.4). p A measure as in the above statement satisfies, in particular, the con- dition in the following one and shows that, in general,d(m)6ds(m).

EXAMPLE4.3. There exist probability measuresmsuch that d_s(m)ˆ0 (minimum possible value) andd(m)ˆN (maximum possible value).

We have stated in Section 1 that the information that, for a probability measuremand a real numbera2]0;1[, the critical dimensiondacoincides withthe irrigability dimensiond(m), (i.e.aˆ 1

(d(m))⁰) does not allow to de- cide whether the measure is irrigable or not. Examples 4.4 and 4.5 will motivate this claim.

EXAMPLE4.4. Letmbe an Ahlfors probability measure in dimension b0. Thenmis nota-irrigable ifd_aˆbˆd(m).

PROOF. The thesis follows from Corollary 1.6. p REMARK 4.2. One has Ahlfors regular measures for every dimension b<N. Indeed, letCbe a selfsimilar (Cantor) set of R^N with dimension b>0. Let us call H^b_bC the Hausdorff measure distributed on C, i.e. the measure onR^Ndefined setting8XR^N

H^b_bC(X)ˆ H^b(X\ C): ThenH^b_bC is Ahlfors regular with dimensionb.

EXAMPLE4.5. There exist some measuresmfor whichd(m)is a mini- mum, i.e. there exist some measuresmand some exponentsa2]0;1[such thatd(m)ˆda andmisa-irrigable.

PROOF. Indeed, let us fixa2]0;1[ and let (C_n)_n2Nbe a sequence of self similar (Cantor) sets inR^Nwithdimensiondanwhere (an)_n2Nis a sequence

converging toafrom below, by Corollary 1.5 and Remark 4.2 we know that d(H^d_bC^an_n)ˆdan<daand, by Remark 1.3 (1), we get thatH^d_bC^an_nisa-irrigable.

Let us consider a suitable sequence (en)_n2Nof positive real numbers, suf- ficiently small to allow us to considermˆ P

n2Ne_nH^d_bC^an_n. We know, by Cor- ollary 4.2 that alsoe_nH^d_bC^an_nare irrigable and we callx_nan irrigation pattern which irrigatesenH^d_bC^an_n(i.e. suchthatIa(x_n)<‡ 1andm_xnˆenH^d_bC^an_n). Under the choice of a sufficiently infinitesimal sequence of coefficients (en)_n2N, we have P

n2NIa(x_n)<‡ 1.

Now let us consider the bunch x of the sequence of patterns x_n (see (A.1)) so that by Remark A.1 we have

m_xˆm …4:5†

and

I_a(x)X

n2N

I_a(x_n)<‡ 1: …4:6†

Equality (4.5) and inequality (4.6) give the a-irrigability of m and therefore d(m)da. Moreover d(m)da. Indeed being, for all n2N, menH^d_bC^an_n we get, by Corollary 4.1 and Remark 4.2 that d(m)d(enH^d_bC^an_n)ˆd(H^d_bC^an_n)ˆdan!da. p

5. Discretizations and resolution dimensions of a measure.

DEFINITION 5.1. We shall say that a measure m is a discrete measure if

card( supp (m))<1 and we shall callcard( supp (m))``resolution'' ofm.

DEFINITION5.2. For every n2N we shall denote by D_n the set con- taining all the discrete probability measures whose resolution is less or equal to n. Equivalently, Dnis the set of all the convex combinations of n Dirac masses.

For any p1 we recall the definition of Kantorovitch-Wasserstein distance of indexp.

DEFINITION5.3. Let p1 and letm, nbe two probability measures.

We define the Kantorovitch-Wasserstein distance of index p between m andn by

d_p(m;n)ˆ min

s

Z

VV

jx yj^pds 0

@

1 A

1p

;

where the minimum is taken on all the transport plansswhich leadmton, i.e. measures onVVsuch that their push forward measures by the first and the second projection on V respectively are mand n (p_1#sˆmand p_2#sˆn) (see[1] for more details).

DEFINITION5.4. Letmbe a probability measure. For every n2N, given p1, we shall denote bym_n(or, when necessary, bym^pn) one of the elements of D_n of minimal distance with respect to the Kantorovitch-Wasserstein distance of index p fromm. We shall refer to m_n as to a discretization of resolution n ofm(with respect to the index p).

PROPOSITION5.1. Let1pq and let m,n be two probability mea- sures. Then

d_p(m;n)d_q(m;n) …5:1†

and

dq(m;n)d^{1 pq} dp(m;n)^p

q; …5:2†

where the constant d is the diameter ofsupp (m)[supp (n).

PROOF. Lettbe an optimal transport plan frommtonwithrespect to the Kantorovitch-Wasserstein distance of index q. Then, by HoÈlder in- equality,

[dp(m;n)]^p Z

VV

jx yj^pdt Z

VV

jx yj^qdt 0

@

1 A

pq Z

VV

dt 0

@

1 A

1 ^p_q

ˆ[dq(m;n)]^p:

To prove (5.2) we shall consider an optimal transport plantfrommton withrespect to thepdistance. Let us calldˆ( supp (m)[supp (n)), then

[d_q(m;n)]^q Z

VV

jx yj^qdtd^{q p} Z

VV

jx yj^pdtˆd^{q p}[d_p(m;n)]^p;

from which the thesis follows. p

In the following, for any n2N and p1, we shall use the Kantor- ovitch-Wasserstein distance of indexpofmfromDn

d^p_nˆd_p(m;m_n)ˆd_p(m;D_n); …5:3†

to test ``how good'' a discretization of resolutionncan be. When we shall deal with more than one measure we shall use the more detailed notation d^p_n(m)ˆdp(m;Dn).

In [3] the following proposition, which gives a relation between the cost of an irrigation patternxand the Kantorovitch-Wasserstein distanced^1a₁(m_x) of the irrigation measurem_x from a Dirac delta, has been proved.

PROPOSITION5.2. Letxbe an irrigation pattern, with a source point S, then

d^1a₁(m_x)d¹_a(m_x;dS)Ia(x):

REMARK 5.1. Let 1pq, n2N, n1 and letm be a probability measure. Then from(5.1)and(5.2),applied fornˆm^pnandnˆm^qn, we get

d^p_nd^q_n …5:4†

and

d^q_n d^{1 pq} d^p_n^p_q

; …5:5†

where the constant d is the diameter ofsupp (m).

It is clear that, increasing the numbern, the discretizationsm_nbecome more accurate, therefore it is rather natural to make some decay hy- pothesis ond^p_n.

DEFINITION5.5. Letmbe a probability measure and p1, then we shall callresolution dimensionofmof index p the constant d^pr(m)defined as fol- lows

d^p_r(m)ˆ lim sup

n!‡ 1 log_nd^p_n

₁

: …5:6†

REMARK5.2. Let0<a<(d^pr(m)) ¹, then there exists n such that 8nn : d^p_nn ^a:

Conversely, if a>(d^pr(m)) ¹then for any C>0we have d^p_n>n ^a

for arbitrarily large values of n2N.

Proposition 5.1 allows us to state the corresponding properties ofd^p_r(m) in terms of the index p.

PROPOSITION5.3. Let1pq andma probability measure, then d^p_r(m)d^q_r(m)

…5:7†

and

d^q_r(m)q pd^p_r(m): …5:8†

PROOF. Taking into account (5.6), bothinequalities easily follow from

(5.4) and (5.5). p

REMARK 5.3. It is useful to remark that, by (5.7) and (5.8), d^pr(m) changes with continuity with respect to the index p. Moreover, if there exists a index p1 for which d^p_r(m)ˆ0, then for all q<‡ 1d^q_r(m)ˆ0.

This means that, in such a case, we can not change the resolution di- mension ofmacting on the index p as far as it is finite.

In Appendix B we shall prove the following propositions.

PROPOSITION5.4. Letmbe a probability measure, then d¹_r (m)ˆd_M(m):

PROPOSITION5.5. Letmbe a probability measure. Then d_c(m)d¹_r(m)d^p_r(m) 8p1:

REMARK 5.4. Since in the case pˆ ‡ 1 the dimension d^p_r(m) agrees with dM(m)we shall use the notation d¹_r in order to denote a weaker case, according toProposition 5.3, of the dimension of index‡ 1, defined as

d¹_r (m)ˆsup

p1d^p_r(m):

Example B.1 will show how for pˆ ‡ 1the ``strong'' and the ``weak'' di- mensionsd_M(m) andd¹_r (m) are, in general, distinct.

By the following lemmas we shall estimate the resulution dimension d^pr(m) in the case in which the probability measuremenjoys some regularity properties, beginning by considering a probability measure which satisfies the lower Ahlfors regularity (LAR). As a consequence of Corollary 1.3, taking into account that, by Proposition 5.4, d_M(m)ˆd¹_r (m) we have the following corollary.

COROLLARY5.1. Letmbe a probability measure which satisfies (LAR) in dimensionb0.

Then

d^p_r(m)b 8p1: …5:9†

PROOF. Indeed, by Corollary 1.3, we have d¹_r (m)ˆsup

p1d^pr(m)

dM(m)b. p

In the case in which a probability measuremsatisfies the upper Ahlfors regularity (UAR), by Corollary 1.4 Proposition 5.5 admits the following corollary.

COROLLARY5.2. Letmbe a probability measure such that (UAR) holds true. Then

bd¹_r(m): …5:10†

From Corollary 5.1 and Corollary 5.2 we easily get the following pro- position.

PROPOSITION5.6. Letmbe an Ahlfors regular probability measure of dimensionb0. Then

d^p_r(m)ˆb 8p1: …5:11†

REMARK5.5. So, when the measure is Ahlfors regular, the value of the resolution dimensions s does not depend on the index, while for a generic measure, as it will be shown in the Appendix B by some examples, the resolution dimension is ``out of focus'' in the sense that different indexes give different values. We shall show that, in any case, it is always possible to find an index, suitably characterized, which gives a resolution di- mension which coincides with the irrigability dimension of the measure (seeTheorem 8.1below).