construction of positive definite cubature formulae and approximation of functions via voronoi...

Adv Comput Math (2010) 32:25–41DOI 10.1007/s10444-008-9080-9

Construction of positive definite cubatureformulae and approximation of functionsvia Voronoi tessellations

Allal Guessab · Gerhard Schmeisser

Received: 18 December 2006 / Accepted: 10 April 2008 /Published online: 21 May 2008© Springer Science + Business Media, LLC 2008

Abstract Let � ⊂ Rd be a compact convex set of positive measure. In a

recent paper, we established a definiteness theory for cubature formulae oforder two on �. Here we study extremal properties of those positive definiteformulae that can be generated by a centroidal Voronoi tessellation of �. Inthis connection we come across a class of operators of the form Ln[ f ](x) :=∑n

i=1 φi(x)( f (yi) + 〈∇ f (yi), x − yi〉), where y1, . . . , yn are distinct points in� and {φ1, . . . , φn} is a partition of unity on �. We present best possiblepointwise error estimates and describe operators Ln with a smallest constantin an Lp error estimate for 1 ≤ p < ∞. For a generalization, we introduce anew type of Voronoi tessellation in terms of a twice continuously differentiableand strictly convex function f . It allows us to describe a best operator Ln forapproximating f by Ln[ f ] with respect to the Lp norm.

Keywords Cubature formulae · Positive definite formulae ·Multivariate approximation · Partitions of unity ·Centroidal Voronoi tessellation · Generalized Voronoi tessellation

Mathematics Subject Classifications (2000) Primary 05B45 · 41A63 · 41A80 ·52C22 · 65D30 · 65D32 · Secondary 06A06 · 26B25 · 26D15 · 52A40

Communicated by Tomas Sauer.

A. GuessabDepartment of Applied Mathematics, University of Pau, 64000 Pau, Francee-mail: [email protected]

G. Schmeisser (B)Department of Mathematics, University of Erlangen-Nuremberg, 91054 Erlangen, Germanye-mail: [email protected]

26 A. Guessab, G. Schmeisser

1 Introduction

Let us first fix our terminology. We say that � ⊂ Rd is measurable if it has a

finite Lebesgue measure, which we denote by |�|. For measurable �, the classL1(�) comprises all Lebesgue integrable functions f : � → R. A propertyholds almost everywhere (abbreviated by a. e.) on � if it holds on � except fora set of measure zero.

By C(�), we denote the class of all real-valued continuous functions on� and by Ck(�), where k ∈ N, the subclass of all functions which are k timescontinuously differentiable. As usual, differentiability at boundary points isdefined with the help of directional derivatives. It is convenient to agree thatC0(�) := C(�).

By ‖ · ‖ we mean the euclidean norm in Rd and by 〈x, y〉 the standard inner

product of x, y ∈ Rd.

For f ∈ C2(�), we denote by

H[ f ](x) :=(

∂2 f∂xi∂x j

(x)

)

i, j=1,...,d

the Hessian matrix of f at x and introduce

⎪⎪⎪⎪D2 f⎪⎪⎪⎪ := sup

x∈�

sup{ ∣∣y� H[ f ](x)y

∣∣ : y ∈ R

d, ‖y‖ = 1},

where y is assumed to be a column vector so that its transpose y� becomes arow vector.

From now on let � ⊂ Rd be a compact convex set. The cubature formulae

considered in this paper can be introduced as follows.

Definition 1.1 For n points x1, . . . , xn ∈ �, called nodes, and associatedpositive numbers A1, . . . , An, we say that

{(Ai, xi) : i = 1, . . . , n

}(1)

is a pd-system defining the positive definite cubature formula (or the pd-formula, for short)

∫

�

f (x) dx =n∑

i=1

Ai f (xi) + Rn[ f ] (2)

if Rn[ f ] ≥ 0 for all convex functions f ∈ C(�). A pd-system with distinctpoints x1, . . . , xn is said to be of length n.

It is easily seen that every pd-formula is of linear precision, that is, itsremainder Rn[ f ] vanishes for all affine functions f .

Positive definite cubature 27

In the univariate case with � = [a, b ] and nodes ordered as

a ≤ x1 < · · · < xn ≤ b ,

the remainder of a quadrature formula of linear precision has a representation

Rn[ f ] =∫ b

aK(x) f ′′(x) dx

for f ∈ C2[a, b ], where

K(x) := (x − a)2

2−

n∑

i=1

Ai(x − xi)+

is the second Peano kernel and x+ := (x + |x|)/2; also see [1, Section II.4] or[2, Section 4.3]. On all compact subsets of (a, b), a convex function f ∈ C[a, b ]can be approximated arbitrarily close by twice continuously differentiablefunctions with a non-negative second derivative. Therefore it is easily seen thatthe pd-formulae on [a, b ] are exactly those quadrature formulae whose secondPeano kernel K is non-negative on [a, b ] and

∑ni=1 Ai = b − a, K(b) = 0. The

last two equations guarantee linear precision.A detailed study of pd-formulae can be found in [9]. We now recall some of

those considerations.For two pd-systems

a := {(Ai, xi) : i = 1, . . . , n

}and b := {

(B j, y j) : j = 1, . . . , m},

we write a ≺ b ifn∑

i=1

Ai f (xi) ≤m∑

j=1

B j f (y j)

for every convex f ∈ C(�). The binary relation ≺ defines a preorder on thecollection of all pd-systems. We say that a pd-system a of length n defines amaximal pd-formula if there is no pd-system b of length n such that a ≺ b anda �= b.

For every f ∈ C2(�), we have the sharp error estimate [9, Proposition 2.1]

|Rn[ f ]| ≤ Rn[‖ · ‖2] ·

⎪⎪⎪⎪D2 f⎪⎪⎪⎪

2. (3)

We say that a pd-formula is optimal if among all pd-formulae with n nodes,Rn[‖ · ‖2] attains a smallest value. It was observed in [9, Remark 5.2] that eachoptimal pd-formula is maximal.

For an investigation of pd-formulae, a correspondence between these for-mulae and partitions of unity is very useful. To be precise, we say that a system{φ1, . . . , φn} of real-valued functions is a partition of unity on � if:

(1) φi ∈ L1(�) and∫�

φi(x) dx > 0 for i = 1, . . . , n;(2) φi(x) ≥ 0 a. e. on � for i = 1, . . . , n;(3) φ1(x) + · · · + φn(x) = 1 a. e. on �.


One of the main results on pd-formulae is the following theorem [9,Theorem 3.4, Theorem 3.8].

Theorem A Let � ⊂ Rd be a compact convex set of positive measure. Then

{(Ai, xi) : i = 1, . . . , n} defines a positive definite cubature formula on � if andonly if there exists a partition of unity {φ1, . . . , φn} on � such that

Ai :=∫

�

φi(x) dx and xi := 1

Ai

∫

�

xφi(x) dx (i = 1, . . . , n).

In view of this theorem, we say that partitions of unity generate pd-systems andpd-formulae.

A system {�1, . . . , �n} of subsets of � is called a decomposition of � if:

(1) �i is measurable and |�i| > 0 for i = 1, . . . , n;(2)

∣∣�i ∩ � j

∣∣ = 0 if i �= j;

(3) �1 ∪ · · · ∪ �n = �.

Every decomposition of � induces a partition of unity {φ1, . . . , φn} given bythe characteristic functions

φi(x) = 1�i(x) :={

1 if x ∈ �i

0 if x ∈ � \ �i(i = 1, . . . , n).

It is desirable to have a simple and suggestive name for this type of partitionsof unity.

Definition 1.2 A partition of unity {φ1, . . . , φn} is said to be a decomposition of� if there exists a decomposition {�1, . . . , �n} of � such that

φi(x) = 1�i(x) a.e. on �

for i = 1, . . . , n.

A special class of decompositions of � are the Voronoi tessellations. Givendistinct y1, . . . , yn ∈ �, the Voronoi sets generated by these points are definedby

�i :={

x ∈ � : ‖x − yi‖ ≤ ‖x − y j‖, j = 1, . . . , n}

(4)

for i = 1, . . . , n. These sets deviate from the Voronoi regions defined in [3]by sets of measure zero only, which has no effect on our applications. Thesystem of Voronoi sets {�1, . . . , �n} is a decomposition of �, which is called aVoronoi tessellation. It generates a pd-system {(|�i| , xi) : i = 1, . . . , n}, where,in general, xi �= yi, that is, the generating points of the Voronoi tessellationand the nodes of the associated pd-formula need not be the same.

A Voronoi tessellation {�1, . . . , �n} generated by y1, . . . , yn is said to becentroidal if

yi = 1

|�i|∫

�i

x dx.


In this case, the generated pd-system is {(|�i| , yi) : i = 1, . . . , n}, which showsthat now the generating points of the Voronoi tessellation coincide with thenodes of the associated pd-formula.

Centroidal Voronoi tessellations are of relevance for extremal problems indiverse branches of mathematics, science and engineering; see [3, Section 2].Intuitively, one may think that they will also generate “good” pd-formulae. Weshall support this suggestion by demonstrating an extremal property of thesepd-formulae; see Theorems 2.1 and 2.2 below.

It should be mentioned that there exist efficient algorithms for the com-putation of centroidal Voronoi tessellations of a domain; see [3, 4]. Hence theproblem of finding good pd-formulae, which was briefly discussed in [9, Section5], has a satisfactory solution.

In [9, Section 6] it was shown that all pd-formulae can also be obtained asfollows. Let � = {φ1, . . . , φn} be a partition of unity and let y1, . . . , yn ∈ �.

Define

Ln[ f ](x) :=n∑

i=1

φi(x)( f (yi) + 〈∇ f (yi), x − yi〉

)(5)

and integrate Ln[ f ] over �. This leads to a cubature formula that involves thevalues of f and those of the gradient ∇ f at the points y1, . . . , yn. It turns outthat the points yi can be chosen such that the coefficients of the componentsof ∇ f (yi) vanish for i = 1, . . . , n, and this choice produces the pd-formulagenerated by �.

Here we consider the approximation of f by Ln[ f ]. We look for operatorswithin the class (5) which give an optimal approximation with respect to the Lp

norm (1 ≤ p < ∞) and describe them in terms of special Voronoi tessellations.

2 Statement of results

The following theorem is an immediate consequence of a known extremalproperty of centroidal Voronoi tessellations.

Theorem 2.1 Every optimal positive definite cubature formula can be generatedby a centroidal Voronoi tessellation.

In general, an optimal pd-formula is not uniquely determined. For example,suppose we have an optimal pd-formula on a ball �. Then each rotationof the whole configuration around the center of the ball produces anotheroptimal pd-formula and an infinite number of them are different. Hencethere may be many centroidal Voronoi tessellations that generate optimalpd-formulae. Furthermore, there may be centroidal Voronoi tessellations thatdo not generate an optimal pd-formula, that is, Theorem 2.1 does not have aconverse.

For example, Fig. 1 shows three centroidal Voronoi tessellations of a squarefor n = 2. The first two generate two different optimal pd-formulae while the


�

�

�

�

� �

Fig. 1 Centroidal Voronoi tessellations of a square for n = 2

third one generates a pd-formula that is not optimal. Indeed, the remainderR2[‖ · ‖2] of the third formula exceeds that of the first two formulae bythe factor 16/15. Does the pd-formula generated by any centroidal Voronoitessellation always have a certain extremal property? Here is an answer to thisquestion.

Theorem 2.2 Every centroidal Voronoi tessellation generates a maximal posi-tive definite cubature formula.

Now a new question arises. Can every maximal pd-formula be generated bya centroidal Voronoi tessellation? The following answer will be obtained by anexample.

Proposition 2.3 There exist maximal positive definite cubature formulae thatcannot be generated by a centroidal Voronoi tessellation.

This proposition does not diminish the value of centroidal Voronoi tessel-lations. It rather shows that a maximal pd-formula need not necessarily be a“good” one in the following sense. When we look for an optimal formula, theset of all formulae generated by a centroidal Voronoi tessellation is a betterpreselection than the set of all maximal formulae.

The preceding results and the accompanying considerations can be summa-rized as follows. Denote by S the class of all pd-formulae with n nodes andadd the subscripts opt and max for the subclasses of optimal and maximal pd-formulae, respectively. Write ScV for those pd-formulae that can be generatedby a centroidal Voronoi tessellation. Then

Sopt � ScV � Smax � S .

Possibly there is a big difference between the classes ScV and Smax. Thereare some indications that the following statement may be true.

Conjecture 2.4 Let {�1, . . . , �n} be a decomposition of �. Suppose that thesets �1, . . . , �n are all convex. Then {1�1 , . . . , 1�n} generates a maximal posi-tive definite cubature formula.


The sets �1, . . . , �n of a Voronoi tessellation of � are always convex.Hence, if the conjecture is true, any Voronoi tessellation, no matter whether itis centroidal or not, will give a maximal pd-formula.

The following statement can be easily verified.

Proposition 2.5 In dimension d = 1 the conjecture is true.

In dimension 1, a set is convex if and only if it is connected. Therefore,one might think that the conjecture could already be true for connected sets�1, . . . , �n. However, this is not the case.

Proposition 2.6 In the above conjecture, the property “convex” cannot bereplaced by “connected” when the dimension is greater than 1.

The following proposition is another contribution to the above conjecture.

Proposition 2.7 The conjecture is true in any dimension when n ≤ 2.

We now turn to the operator (5). An error estimate can be deduced fromthe following result, which is an extension of [9, Proposition 2.1] from linearfunctionals to linear operators.

Theorem 2.8 For k ∈ {0, 1, 2}, let A : Ck(�) → L1(�) be a linear operator.The following statements are equivalent:

(1) For every convex function g ∈ C2(�), we have

A[g](x) ≤ g(x) a.e. on �.

(2) For every function f ∈ C2(�), we have

| f (x) − A[ f ](x)| ≤ (‖x‖2 − A[‖ · ‖2

](x)

)⎪⎪⎪⎪D2 f

⎪⎪⎪⎪

2a.e. on �.

Equality is attained for all functions of the form

f (x) := a(x) + c‖x‖2, (6)

where c ∈ R and a(·) is any affine function.

It is easy to see that statement (1) holds when A is the operator Ln definedin (5). Indeed, if g is as in statement (1), then it has for each y ∈ � a tangentialhyperplane which supports its graph from below; also see [12, p. 98, TheoremA]. In particular,

g(yi) + 〈∇g(yi), x − yi〉 ≤ g(x) (x ∈ �, i = 1, . . . , n).

Now, multiplying both sides by φi(x) and summing over i from 1 to n, we findthat

Ln[g](x) ≤ g(x) a.e. on �.


Hence Theorem 2.8 implies the following error estimate.

Corollary 2.9 For every f ∈ C2(�), we have

| f (x) − Ln[ f ](x)| ≤ (‖x‖2 − Ln[‖ · ‖2

](x)

)⎪⎪⎪⎪D2 f

⎪⎪⎪⎪

2a.e. on �. (7)

Equality is attained for all functions of the form (6)

In view of this error estimate, we call the operator Ln in (5) optimalwith respect to the Lp norm if for f (x) := ‖x‖2 the partition of unity � andthe points y1, . . . yn are chosen such that the Lp norm of f − Ln[ f ] on �,defined by

∥∥ f − Ln[ f ]∥∥p,�

:=[∫

�

∣∣ f (x) − Ln[ f ](x)

∣∣p dx

]1/p

,

attains its smallest value denoted by cn,p.Theorem 2.1 says equivalently that if Ln is optimal with respect to the

L1 norm, then there is a centroidal Voronoi tessellation whose characteristicfunctions constitute the partition of unity � and whose generating points arey1, . . . , yn. For general p ∈ [1, ∞), the following result holds.

Theorem 2.10 For p ∈ [1, ∞) there exists an operator Ln of the form (5) whichis optimal with respect to the Lp norm. Denoting by {�1, . . . , �n} the Voronoitessellation generated by y1, . . . , yn, we have φi = 1�i almost everywhere on �

for i = 1, . . . , n and, in addition, the equations

yi

∫

�i

‖x − yi‖2p−2dx =∫

�i

x‖x − yi‖2p−2dx (i = 1, . . . , n) (8)

are satisfied.

Asymptotic results on the error constant cn,p and the distribution of thepoints y1, . . . yn for an optimal operator when n is large can be deduced from ageneral theorem of Gruber [7, Satz 4], [8, Theorem 2] by employing his resultfor f (t) = t2p. In particular, it follows that n2/dcn,p approaches a positive limitas n → ∞.

In dimension d = 2, asymptotic results on the centroidal Voronoi tessella-tion figuring in Theorem 2.1 (or in the case p = 1 of Theorem 2.10) have alsobeen obtained by various other authors; see, e.g., [5, 6, 11]. For large n, theVoronoi sets of the centroidal tesselation are approximately congruent regularhexagons.

When L̃n is optimal, then, by definition, L̃n[‖ · ‖2] is a best approximationto ‖ · ‖2 among all approximations Ln[‖ · ‖2] with Ln of the form (5). However,for f �= ‖ · ‖2, there may exist an operator Ln such that Ln[ f ] is a betterapproximation to f than L̃n[ f ]. When f is twice continuously differentiableand strictly convex, a best operator Ln for approximating f by Ln[ f ] can bedescribed by a generalization of the Voronoi tessellation.


Consider

qf (x, yj) := f (x) − f (yj) − ⟨∇ f (yj), x − yj⟩.

Under our assumptions on f , the function qf (·, yj) is non-negative, strictly con-vex and, together with its gradient, it vanishes at yj. Furthermore, qf (·, yi) −qf (·, yj) is an affine function. Now the idea is to let qf (x, yj) take the role of‖x − yj‖2 in (4). Defining

� f,i := {x ∈ � : qf (x, yi) ≤ qf (x, yj), j = 1, . . . , n

}

for i = 1, . . . , n, we easily see that these sets are convex, no two of themhave common interior points, and their union is �. We call {� f,1, . . . , � f,n}a generalized Voronoi tessellation with respect to f , generated by y1, . . . , yn.

When f = ‖ · ‖2, we recover the classical Voronoi tessellation.

Theorem 2.11 Let f ∈ C 2(�) be strictly convex. Then, for p ∈ [1, ∞), thereexists an operator Ln of the form (5) for which Ln[ f ] is a best approximationto f with respect to the Lp norm. Denoting by {� f,1, . . . , � f,n} the generalizedVoronoi tessellation with respect to f generated by y1, . . . , yn, we have φi = 1� f,i

almost everywhere on � and, in addition, the equations

H[ f (yi)]yi

∫

� f,i

(qf (x, yi)

)p−1 dx =∫

� f,i

H[ f (yi)]x(qf (x, yi)

)p−1 dx

are satisfied for i = 1, . . . , n.

This result can be proved by a straightforward modification of the proof ofTheorem 2.10. The latter proof is given at the end of the subsequent section.

3 Proofs

Theorem A shows that there exists a surjective mapping F that maps thecollection of all partitions of unity on � onto the collection of all pd-systems.However, this mapping is not injective. Thus {F−1(a)} denotes the set of allpartitions of unity which generate a.

If {φ1, . . . , φn} and {ψ1, . . . , ψn} belong to {F−1(a)}, then there exists apermutation π of the numbers 1, . . . , n such that

∫

�

φi(x)dx =∫

�

ψπ(i)(x)dx and∫

�

xφi(x)dx =∫

�

xψπ(i)(x)dx

for i = 1, . . . , n. For sake of simplicity, we want to agree that the elements ofany two members of {F−1(a)} are always numbered such that π is the identity.


Proof of Theorem 2.1 Let a = {(Ai, xi) : i = 1, . . . , n} be a pd-system definingan optimal pd-formula. Then, by [9, Theorem 5.1], the set {F−1(a)} containsonly decompositions of �. Let {1�1, . . . , 1�n} be one of them. By Theorem A,

∫

�

1�i(x)〈x, xi〉dx = Ai‖xi‖2 =∫

�

1�i(x)‖xi‖2dx (i = 1, . . . , n).

Hence

Rn[‖ · ‖2] =

∫

�

‖x‖2dx −n∑

i=1

Ai‖xi‖2

=n∑

i=1

∫

�

1�i(x)(‖x‖2 − 2〈x, xi〉 + ‖xi‖2

)dx

=n∑

i=1

∫

�i

‖x − xi‖2dx.

By [3, Proposition 3.1], a necessary condition for minimizing the last expres-sion is that {�1, . . . , �n} is a centroidal Voronoi tessellation generated byx1, . . . , xn. This completes the proof. ��

In the three subsequent proofs, we shall use a result in [9, Theorem 4.8],which we state as a lemma.

Lemma 3.1 Let a be a pd-system of length n. If all members of {F−1(a)}are decompositions of �, then a defines a maximal positive definite cubatureformula.

Proof of Theorem 2.2 Let {�1, . . . , �n} be a centroidal Voronoi tessellationgenerated by x1, . . . , xn and let a = {(Ai, xi) : i = 1, . . . , n} be the pd-systemgenerated by �∗ := {1�1, . . . , 1�n}.

For arbitrary partitions of unity � = {φ1, . . . , φn}, we introduce the func-tional

E(�) :=n∑

i=1

∫

�

φi(x)‖x − xi‖2dx.

Recalling Theorem A, we find by a short calculation that

E(�) =∫

�

‖x‖2dx −n∑

i=1

Ai‖xi‖2 = E(�∗) for � ∈ {F−1(a)}. (9)

Now, assume that {F−1(a)} contains a partition of unity � = {φ1, . . . , φn}which is not a decomposition of �. Then, defining

�ij := {x ∈ � : φi(x) > 0

} ∩ � j , (10)


we conclude that the sets �ij for i �= j cannot be all of measure zero.Next, define

φ̃i := φi −n∑

j=1j�=i

1�ijφi +n∑

j=1j�=i

1� jiφ j (i = 1, . . . , n)

and

�̃i := �i \(⋃

j�=i

� j

)

(i = 1, . . . , n).

These sets are pairwise disjoint and their union differs from � by a set ofmeasure zero only. For x ∈ �̃i, we have 1�ij(x) = 0 for j �= i, and so

φ̃i(x) = φi(x) +n∑

j=1j�=i

1� ji(x)φ j(x) ≤n∑

j=1

φ j(x) ,

which shows that

0 ≤ φ̃i(x) ≤ 1 a.e. on �̃i . (11)

If x ∈ �̃k, then x �∈ �i for every i �= k. Therefore x �∈ � ji for every i �= k andany j ∈ {1, . . . , n}. Hence

φ̃i(x) = φi(x) − 1�ik(x)φi(x) = 0 for x ∈ �̃k, k �= i. (12)

Furthermore, writing∑

i �= j for the summation over all pairs i, j such that i, j ∈{1, . . . , n} and i �= j, we find that

n∑

i=1

φ̃i =n∑

i=1

φi −n∑

i=1

n∑

j=1j�=i

1�ijφi +n∑

i=1

n∑

j=1j�=i

1� jiφ j

= 1 −∑

i �= j

1�ijφi +∑

i �= j

1� jiφ j = 1 a.e. on �. (13)

From (11)–(13), we conclude that �̃ := {φ̃1, . . . , φ̃n} is a partition of unitysuch that φ̃i(x) = 1�i(x) almost everywhere on �. Hence, �̃ ∈ {F−1(a)}, whichimplies together with (9) that

E(�̃) = E(�∗) = E(�). (14)


On the other hand,

E(�̃) =n∑

i=1

∫

�

φ̃i(x)‖x − xi‖2dx

= E(�) −∑

i �= j

∫

�ij

φi(x)‖x − xi‖2dx +∑

i �= j

∫

� ji

φ j(x)‖x − xi‖2dx

= E(�) −∑

i �= j

∫

�ij

φi(x)(‖x − xi‖2 − ‖x − x j‖2

)dx.

Since �ij ⊆ � j, it follows from the definition of Voronoi sets that

‖x − xi‖2 > ‖x − x j‖2 a.e. on �ij.

Now, recalling that at least one of the sets �ij for i �= j has positive measure,we conclude using a standard result in measure theory (see, e.g., [10, p. 104,Theorem D]) that E(�̃) < E(�). This contradicts (14). Hence, all members of{F−1(a)} are decompositions of �. By Lemma 3.1, this implies that a defines amaximal pd-formula. ��

Proof of Proposition 2.3 Let � be the interval [0, 4], and define

φ1(x) ={

1 if x ∈ [0, 1],0 if x ∈ (1, 4].

Then {φ1, 1 − φ1} is a partition of unity on � which generates the pd-systema := {(1, 1

2 ), (3, 52 )}.

Assume that {F−1(a)} contains a partition of unity {ψ1, ψ2} which is not adecomposition of �. Then 1 − ψ1 is positive on a subset of positive measure in[0, 1] and ψ1 is positive on a subset of positive measure in [1, 4]. Therefore,

0 =∫ 4

0xφ1(x) dx −

∫ 4

0xψ1(x) dx =

∫ 1

0x(1 − ψ1(x)

)dx −

∫ 4

1xψ1(x) dx

<

∫ 1

0

(1 − ψ1(x)

)dx −

∫ 4

1ψ1(x) dx =

∫ 4

0φ1(x) dx −

∫ 4

0ψ1(x) dx = 0.

This is a contradiction. Hence, by Lemma 3.1, a defines a maximal pd-formula.However, there is no centroidal Voronoi tessellation of [0, 4] with 1

2 and 52 as

generating points. ��

Proof of Proposition 2.5 Let {�1, . . . , �n} be a decomposition of � into con-vex sets. In the one-dimensional case, this means that � is an interval [a, b ],and we have subintervals �i with end-points ti−1 and ti for i = 1, . . . , n suchthat

a = t0 < t1 < · · · < tn−1 < tn = b .


The characteristic functions of the subintervals �i generate the pd-system

a = { (ti − ti−1 , (ti + ti−1)/2

) : i = 1, . . . , n}.

As an intermediate result, we shall prove by induction on n that if

{ψ1, . . . , ψn} ∈ {F−1(a)},then ψi = 1�i almost everywhere on � for i = 1, . . . , n.

This statement is trivial when n = 1. Now, let n ≥ 2. If {ψ1, . . . , ψn} gener-ates a, then, in particular,

∫ b

aψn(x)dx = tn − tn−1 and

1

tn − tn−1

∫ b

axψn(x)dx = tn + tn−1

2.

Similarly to the previous proof, we see that these equations cannot hold ifψn were positive on a subset of positive measure in [a, tn−1]. Therefore, weconclude that

ψn(x) = 1�n(x) a.e. on [a, b ]. (15)

As a consequence, we obtain

ψ1(x) = · · · = ψn−1(x) = 0 a.e. on [tn−1, b ]. (16)

Now, the restrictions to [a, tn−1] of the functions ψ1, . . . , ψn−1 form a partitionof unity on [a, tn−1] that generates on this interval the same pd-formula as{1�1 , . . . , 1�n−1}. Using the induction hypothesis, we conclude that

ψi(x) = 1�i(x) a.e. on [a, tn−1] for i = 1, . . . , n − 1. (17)

Combining (15)–(17), we complete the proof of the desired intermediateresult.

In particular, we have shown in the univariate case that under the hypothesisof the conjecture, all members of {F−1(a)} are decompositions of �. Hence, byLemma 3.1, a defines a maximal pd-formula. ��

Proof of Proposition 2.6 Let � be the d-dimensional rectangle [0, 2] ×[−1/2, 1/2]d−1. In the (x1, x2)-plane decompose [0, 2] × [−1/2, 1/2] into con-nected subsets G1 and G2 as shown in Fig. 2, and set

�1 := G1 ×[

−1

2,

1

2

]d−2

, �2 := G2 ×[

−1

2,

1

2

]d−2

.

Then {�1, �2} is a decomposition of � into connected subsets. It generates thepd-system

a ={(

1,

(17

32, 0, . . . , 0

))

,

(

1,

(47

32, 0, . . . , 0

))}

.


01.2510.75 2

-0.50

-0.25

0.25

0.50

x1

x2

G 1 G 2

�

�

Fig. 2 Decomposition of a rectangle into two connected sets

Now, for x = (x1, . . . , xd), define

φ1(x) :=

⎧⎪⎪⎨

⎪⎪⎩

31

32if x ∈ � and x1 ∈ [0, 1]

1

32if x ∈ � and x1 ∈ (1, 2]

and φ2 := 1 − φ1. Then � := {φ1, φ2} is a partition of unity on �. A simplecalculation shows that � also generates a, that is, � ∈ {F−1(a)}. Since φ1(x) ≥1/32 for all x ∈ �, a criterion in [9, Theorem 4.9] tells us that a does not definea maximal pd-formula. This completes the proof. ��

Proof of Proposition 2.7 Here, only the case n = 2 is non-trivial. Let {�1, �2}be a decomposition of � into two convex subsets, and denote by a ={(A1, x1), (A2, x2)} the pd-system generated by {1�1 , 1�2}. Clearly, the com-mon boundary of �1 and �2 is part of a hyperplane H. It satisfies an equation

h(x) ≡ h0 + h1x1 + · · · hdxd = 0,

which may be normalized such that h(x1) = 1.

Now, assume that there exists a pd-system b = {(B1, y1), (B2, y2)} such thata ≺ b and a �= b. Then x1 and x2 lie on the line segment that connects y1 andy2; see [9, Theorem 4.6]. For an appropriate numbering of the two elementsof b, we have yi ∈ �i and

yi = x1 + ti(x2 − x1) (i = 1, 2),

where t1 ≤ 0 and t2 ≥ 1. Since a �= b, at least one of these inequalities is strict.Without loss of generality, we may suppose that t1 < 0.


Next, we recall that for every convex function f ∈ C(�), we have

A1 f (x1) + A2 f (x2) ≤ B1 f (y1) + B2 f (y2) ≤∫

�

f (x) dx. (18)

Now define

g(x) :={

h(x) if x ∈ �1 ,

0 if x ∈ � \ �1 .

This is a continuous, convex function on �. Furthermore, {(A1, x1)} and{(A2, x2)} are pd-systems of length 1 on the closures of �1 and �2, respectively.Since pd-formulae are exact for affine functions, we have

∫

�1

g(x) dx = A1g(x1) = A1h(x1) and∫

�2

g(x) dx = A2g(x2) = 0 .

It follows that equality occurs in (18) throughout when f is replaced by g. Inparticular,

A1h(x1) = B1h(y1) . (19)

Since y1 has a bigger distance from H than x1, we have h(y1) > h(x1) = 1, andso (19) implies that A1 > B1.

By interchanging the roles of �1 and �2, and noting that t2 ≥ 1, we findanalogously that A2 ≥ B2. Thus

A1 + A2 > B1 + B2,

which is a contradiction since (18) is valid for f (x) ≡ 1. Hence a defines amaximal pd-formula. ��

Proof of Theorem 2.8 The proof is essentially the same as that of [9,Proposition 2.1]. We therefore do not present all the details.

Let f ∈ C2(�), and suppose that (1) holds. It is easily verified by consideringtheir Hessian matrices that the two functions

g± := ‖ · ‖2

⎪⎪⎪⎪D2 f⎪⎪⎪⎪

2± f

are both convex. Therefore statement (1) implies that

A[‖ · ‖2

]⎪⎪⎪⎪D2 f

⎪⎪⎪⎪

2± A[ f ] ≤ ‖ · ‖2

⎪⎪⎪⎪D2 f⎪⎪⎪⎪

2± f a.e. on �,

which gives the error estimate of statement (2). The case of equality is easilyverified.


Conversely, let g ∈ C2(�) be a convex function, and suppose that statement(2) holds. Define

f := ‖ · ‖2

⎪⎪⎪⎪D2g⎪⎪⎪⎪

2− g

and write E := I − A, where I is the identity on C2(�), for the operator whichproduces the error. It can be shown that

⎪⎪⎪⎪D2 f⎪⎪⎪⎪ ≤ ⎪⎪⎪⎪D2g

⎪⎪⎪⎪. Hence, the errorestimate of statement (2), applied to f , implies that

E

[

‖ · ‖2

⎪⎪⎪⎪D2g⎪⎪⎪⎪

2− g

]

≤ E[‖ · ‖2

]⎪⎪⎪⎪D2g

⎪⎪⎪⎪

2a.e. on �.

This shows that E[g] ≥ 0 almost everywhere on �, as was to be proved. ��

Proof of Theorem 2.10 For f (x) = ‖x‖2, we find by a simple calculation that

f (yi) + 〈∇ f (yi), x − yi〉 = ‖x‖2 − ‖x − yi‖2.

Hence,

‖x‖2 − Ln[‖ · ‖2

](x) =

n∑

i=1

φi(x)‖x − yi‖2 =: [�](x).

Next, let {�1, . . . , �n} be the Voronoi tessellation generated by the pointsy1, . . . , yn. As in the proof of Theorem 2.2, we now consider the sets (10).Assume that there exists a pair k, of different indices such that �k is ofpositive measure. Define the partition of unity �̃ = {φ̃1, . . . , φ̃n} by

φ̃k = φk − 1�kφk , φ̃ = φ + 1�k

φk ,

and φ̃i = φi for i ∈ {1, . . . , n} \ {k, }. Then

[�̃](x) = [�](x) − 1�kφk(x)

(‖x − yk‖2 − ‖x − y‖2).

Recalling the definition of Voronoi sets, we find that

[�̃](x) < [�](x) a.e. on �k

while

[�̃](x) = [�](x) if x ∈ � \ �k.

Consequently, ‖[�̃]‖p,� < ‖[�]‖p,�. From this, we conclude that Ln cannotbe optimal unless φi = 1�i almost everywhere on � for i = 1, . . . , n. In thelatter case,

[�](x) =n∑

i=1

1�i(x)‖x − yi‖2 a.e. on �


and

∥∥[�]∥∥p

p,�=

n∑

i=1

∫

�i

‖x − yi‖2pdx. (20)

Now the existence of an optimal operator Ln is seen by a compactnessargument.

It remains to verify (8). Let v be any element of Rd, and let t ∈ R. In (20),

we replace y j by y j + tv while all the other points yi as well as the sets �1, . . . ,

�n remain untouched. If in (20) the points y1, . . . , yn are those of an optimaloperator Ln, then

g(t) :=∫

� j

∥∥∥x − y j − tv

∥∥∥

2pdx

must have a local minimum at t = 0. Consequently g′(0) = 0, which turns outto be equivalent to

∫

� j

‖x − y j‖2p−2〈x − y j, v〉dx = 0.

Since this must hold for any v ∈ Rd and for each j ∈ {1, . . . , n}, we obtain (8).

��

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