optimum quantization and its applications

http://www.elsevier.com/locate/aim

Advances in Mathematics 186 (2004) 456–497

Optimum quantization and its applications

Peter M. Gruber�

Abteilung fur Analysis, Technische Universitat Wien, Wiedner HauptstraX e 8-10/1142, A-1040 Vienna, Austria

Received 9 October 2002; accepted 11 July 2003

Communicated by Erwin Lutwak

Abstract

Minimum sums of moments or, equivalently, distortion of optimum quantizers play an

important role in several branches of mathematics. Fejes Toth’s inequality for sums of

moments in the plane and Zador’s asymptotic formula for minimum distortion in Euclidean d-

space are the first precise pertinent results in dimension dX2: In this article these results are

generalized in the form of asymptotic formulae for minimum sums of moments, resp.

distortion of optimum quantizers on Riemannian d-manifolds and normed d-spaces. In

addition, we provide geometric and analytic information on the structure of optimum

configurations. Our results are then used to obtain information on

(i) the minimum distortion of high-resolution vector quantization and optimum quantizers,

(ii) the error of best approximation of probability measures by discrete measures and

support sets of best approximating discrete measures,

(iii) the minimum error of numerical integration formulae for classes of Holder continuous

functions and optimum sets of nodes,

(iv) best volume approximation of convex bodies by circumscribed convex polytopes and the

form of best approximating polytopes, and

(v) the minimum isoperimetric quotient of convex polytopes in Minkowski spaces and the

form of the minimizing polytopes.

r 2003 Elsevier Inc. All rights reserved.

MSC: 52C99; 53C20; 94A24; 94A34; 65D30; 65D32; 52A27; 52B60

Keywords: Optimum quantization; Sums of moments; Distortion of vector quantizers; Approximation of

probability measures; Error of numerical integration formulae; Best approximating polytopes; Minimum

isoperimetric quotient

ARTICLE IN PRESS

�Fax: +43-1-558-01-11498.

E-mail address: [email protected].

0001-8708/$ - see front matter r 2003 Elsevier Inc. All rights reserved.

doi:10.1016/j.aim.2003.07.017

1. Introduction

1.1. Fejes Toth’s inequality for sums of moments in the plane

Let f : ½0;þNÞ-½0;þNÞ be increasing, where f ð0Þ ¼ 0; and let H be a convex 3,

4, 5, or 6-gon in the Euclidean plane E2: Then

infSCE2

#S¼n

ZH

minsAS

f f ðjjs ujj2Þg du

� �Xn

ZHn

f ðjjujj2Þ du; ð1:1Þ

where n ¼ #S is the number of points of S and Hn a regular convex hexagon in E2 ofarea jHj=n and center at the origin o: jj � jj2 and j � j stand for the Euclidean norm and

the ordinary (area) measure. There are more than a dozen proofs for this result of L.Fejes Toth [17] and its variants. Its applications range from packing and coveringproblems for solid circles, problems of optimal location in econometrics,

quantization of data, Gauss channels and numerical integration, all in E2; to theisoperimetric problem for special convex polytopes and asymptotic approximation

of convex bodies in E3: See [29,39] for references. Related results, including results onfinite difference methods, cellular biology and territorial behaviour of animals aresurveyed by Du et al. [12].

Inequality (1.1) indicates that for certain geometric and analytic problems in E2 or

E3 regular hexagonal configurations are close to optimal, possibly optimal.

1.2. The corresponding planar stability problem

The objective here is to describe the sets S ¼ Sn for which equality or approximateequality holds in (1.1). Although for certain functions f no clear cut answer is

possible, Gruber [28,30] showed that in E2 and on Riemannian 2-manifolds undersuitable restrictions for f ; the sets Sn form ‘asymptotically regular hexagonal

patterns’ as n-N: For E2 a similar, slightly more general result was given by G.Fejes Toth [15].

The stability result implies that for certain geometric and analytic problems in

E2; E3 and on Riemannian 2-manifolds, the optimal or almost optimal configura-tions are ‘asymptotically regular hexagonal pattern’. For examples see Gruber [30].

1.3. Extensions to general dimensions

From the point of view of applications the following extensions suggestthemselves. First, determine or estimate the expressions

infSCM#S¼n

ZJ

minpAS

f f ðRmðp; xÞÞgwðxÞ doMðxÞ� �

; ð1:2Þ

ARTICLE IN PRESSP.M. Gruber / Advances in Mathematics 186 (2004) 456–497 457

and

infSCEd

#S¼n

ZJ

minsAS

f f ðjjs ujjÞgwðuÞ du

� �: ð1:3Þ

Here J is a (Jordan) measurable set in a Riemannian d-manifold M with

Riemannian metric RM and (area) measure oM ; or in Ed ; where the latter is

endowed with an additional norm jj � jj; and w : J-Rþ a weight function. Second,describe the corresponding minimizing or almost minimizing configurations. Whileprecise answers to these problems are out of reach, asymptotic estimates and upperand lower bounds for the expressions in (1.2) and (1.3) and information on theminimizing configurations have been given for special functions f ; particular norms,and d ¼ 2:

In the context of information theory Zador [48] proved that for f ðtÞ ¼ ta; a40;and jj � jj ¼ jj � jj2 the expression in (1.3) is asymptotically equal to

ZJ

wðuÞd

aþd du

� �aþdd 1

nad

as n-N;

omitting a multiplicative constant; for related, more recent work in informationtheory see the survey of Gray and Neuhoff [21], the book [9] and the articles [22], [23]and [41]. The results in information theory were used to approximate probabilitymeasures by discrete measures, see the report of Graf and Luschgy [20]. In thedisguise of an asymptotic formula for the minimum error of numerical integrationformulae the same result, but for general norms instead of jj � jj2; was proved by

Chernaya [7]. A similar formula with f from a wide class of moduli of continuityincluding the functions of the form ta and where the asymptotics is described by afunction related to f ; was given in the same year by Chernaya [8]. Finally, Gruber[26] and Glasauer and Gruber [19] proved in the context of convex geometry that the

expression in (1.2) for f ðtÞ ¼ t2 is asymptotically equal to

ZJ

wðxÞd

dþ2 doMðxÞ� �dþ2

d 1

n2d

as n-N;

again omitting a multiplicative constant. It is curious to note that all these closelyrelated results were found independently in different areas of mathematics. Theproofs differ, but a crucial idea, a similarity argument, which was communicated toZador by Hammersley, is the same in each of them. In [33] we will describe a prooffor general norms which is also based on it.

If f is strictly increasing, then for S ¼ fsn1;y; snngCEd ;

ZJ

mini¼1;y;n

f f ðjjsni ujj2ÞgwðuÞ du ¼Xn

i¼1

ZDni

f ðjjsni ujj2ÞwðuÞ du;

ARTICLE IN PRESSP.M. Gruber / Advances in Mathematics 186 (2004) 456–497458

where the Dirichlet–Voronoi cells Dni; i ¼ 1;y; n; in J corresponding to S aredefined by

Dni ¼ fuAJ: jjsni ujj2pjjsnj ujj2 for j ¼ 1;y; ng:

(This explains why we speak of sums of moments.) A conjecture of Gersho [18]

asserts that for f ðtÞ ¼ t2 or, more generally, for strictly increasing f ; and for jj � jj ¼jj � jj2 and w ¼ 1; there is a convex polytope P with jPj ¼ 1 which admits a tiling

of Ed by congruent copies such that the following holds: as n-N; the cells Dni

which correspond to minimizing configurations S ¼ Sn; where #Sn ¼ n; are

‘asymptotically congruent’ to ðjJj=nÞ1=dP: For d ¼ 2 the author’s stability result

stated above readily implies the truth of this conjecture, showing also thatP is a regular hexagon. This seems to be the first proof of Gersho’s conjecture ford ¼ 2: Also G. Fejes Toth’s [15] version of the stability theorem yields a proof ofGersho’s conjecture for d ¼ 2: (We point out that no proof is provided by FejesToth [16] or Newman [42], contrary to positive statements in the literature.) FordX3 it is not clear that the conjecture is valid and if so, at present, proofs seem to beout of reach.

Assuming the truth of Gersho’s conjecture, lower bounds for the expression in(1.3) have been given for certain functions f and certain norms. See [33] for simplearguments which also lead to lower bounds. For references see the survey of Grayand Neuhoff [21].

General references for results as described above are Gray and Neuhoff [21], Duet al. [12], Graf and Luschgy [20] and Gruber [33]. The latter is an easilycomprehensible introduction to the present article.

1.4. Results

We will give asymptotic formulae for the expressions in (1.2) and (1.3) for a wideclass of functions f : Further, geometric and analytic information on the minimizingconfigurations S ¼ Sn will be provided; roughly speaking, Sn is distributed over J

rather uniformly and regularly. The hard part of the proof is to show that Sn isDelone (Section 2).

These results admit interpretations and applications to quantization of data,approximation of probability measures by discrete probability measures, numericalintegration, approximation of convex bodies by polytopes and the isoperimetricproblem for convex polytopes.

An important step in data transmission is to encode signals (in many cases vectors

in Ed) produced by a source into codewords (particular vectors) from a codebookconsisting of, say, n codewords (and which are then transmitted in a channel).So-called distortions measure the quality of this encoding or quantizing process.See [21] for the pertinent literature starting with Shannon and Zador. In Shannon’stheory distortion is estimated as d-N; while in Zador’s high resolution theoryestimates for n-N are considered. Our results immediately imply asymptotic


formulae for the minimum distortion of quantization as the number n of codewordstends to infinity and give information about the structure of optimum codebooks(Section 3).

Given a probability measure on a Jordan measurable subset of Ed ; it is a naturalquestion to ask, how well can it be approximated by discrete measures. Here thespace of all probability measures is endowed with the Wasserstein or Kantorovichmetric, or generalizations of it. See Graf and Luschgy [20]. Our results yieldasymptotic formulae for the error of the best approximation as the number ofsupport points of the approximating discrete measures tends to infinity. In addition,information on the structure of the support of the best approximating discretemeasures is provided (Section 4).

Consider a class of Riemann integrable real functions on a Jordan measurable

subset of Ed or of a Riemannian manifold M: In general it is difficult to estimate theminimum error of numerical integration formulae with n nodes and weights.Pertinent results have been given in the context of uniform distribution theory byKoksma and Hlawka, see [34], further for Sobolev spaces of functions by Sobolev[46] and his school, including Polovinkin [43], and for classes of Lipschitz or Holdercontinuous functions by Babenko [1,2] for d ¼ 2 and Sobol’ [45], Chernaya [7,8] andothers for general d: Our results yield for certain Holder classes of functionsasymptotic formulae for the minimum error of numerical integration formulae as thenumber n of nodes and weights tends to infinity. (Our classes are slightly less generalthan those of Chernaya.) Moreover, we obtain information about the structure ofoptimal sets of nodes (Section 5).

The special case f ðtÞ ¼ t2 of our result for Riemannian manifolds yieldsasymptotic formulae for best volume approximation of convex bodies bycircumscribed convex polytopes as the number of facets tends to N: An argumentfrom the proof of our manifold result shows that the facets of the best approximatingpolytopes are all ‘rather round’ and ‘roughly of the same size’ (Section 6).

The latter results can be interpreted as asymptotic formulae for the minimumisoperimetric quotient of convex polytopes in a Minkowski space with n facets asn-N and as information about the form of polytopes with minimum isoperimetricquotient (Section 7).

For notations and results not explained below, see Schneider [44].

2. Sums of moments on Riemannian d-manifolds and normed d-spaces

2.1. Preliminaries

A function f : ½0;þNÞ-½0;þNÞ satisfies the growth condition if

f ð0Þ ¼ 0; f is continuous and strictly increasing and; for any given s41;

the quotient f ðstÞ=f ðtÞ is decreasing and bounded above for t40: ð2:1Þ


In addition to the positive powers of t there are many other functionsf : ½0;þNÞ-½0;þNÞ which satisfy the growth condition. This is shown by thefollowing result. We state it without proof.

Proposition. Let f : ½0;þNÞ-½0;þNÞ be such that f ðtÞ ¼ 0 only for t ¼ 0:Define g : R-R by gðuÞ ¼ log f ðeuÞ for uAR: Then the following statements are

equivalent:

(i) f satisfies the growth condition,(ii) g is strictly increasing, concave and Lipschitz.

Let M be a Riemannian d-manifold. By this we mean a d-dimensional

manifold of class C3 with a metric tensorfield (which has coefficients) of

class C1: Let RM and oM be the corresponding Riemannian metric and (area)measure on M: A set JCM is (Jordan or Riemann) measurable if its closurecl J is compact and oMðbd JÞ ¼ 0; where bd J is the boundary of J: See 2.3for an alternative definition of measurability. A measurable set JCM has positive

density if

there are b41; g40; such that BMðp; RÞ is measurable and

Rd

bpoMðBMðp; RÞ-JÞpbRd for pAcl J; 0oRog; ð2:2Þ

where the ball BMðp; RÞ with center p and radius R is the set fxAM : RMðp; xÞpRg:The measurability of BMðp; RÞ for sufficiently small g follows from an argumentusing the exponential mapping, compare the proof of (2.12). Let JCM bemeasurable with oMðJÞ40; let ðSnÞ be a sequence of sets in M with #Sn ¼ n;

and let d41: Sn is a ð1=dn1=d ; d=n1=dÞ-Delone set in J for n ¼ 1; 2;y; if any two

distinct points of Sn have distance at least 1=dn1=d and for each point of J there is a

point of Sn at distance at most d=n1=d : Sn is uniformly distributed in J as n-N with

respect to an integrable function, a density, w : J-Rþ if for any measurable set KCJ

we have,

#ðK-SnÞBZ

K

wðxÞ doMðxÞZ

J

wðxÞ doMðxÞ� ��

n as n-N:

2.2. Asymptotic results

We will prove the following result, where a40 is a constant corresponding to f ;see (2.14).

Theorem 1. Let f : ½0;þNÞ-½0;þNÞ satisfy the growth condition (2.1) and let a40be the corresponding constant. Then there is a constant div40; depending only on f and


d; such that the following is true: let M be a Riemannian d-manifold, JCM a compact

measurable set with oMðJÞ40; and w : J-Rþ continuous. Then

(i) Fn ¼ infSCM#S¼n

ZJ

minpAS

f f ðRMðp; xÞÞgwðxÞ doMðxÞ� �

Bdiv

ZJ

wðxÞd

aþd doMðxÞ� �aþd

d

f 1

n1d

!as n-N:

If, in addition, J is connected and has positive density, see (2.2), then the following hold:let ðSnÞ be a sequence of sets SnCM with #Sn ¼ n such that the infimum in (i) is

attained for S ¼ Sn; n ¼ 1; 2;y: Then

(ii) there is a constant d41 such that Sn is ð1=dn1=d ; d=n1=dÞ-Delone in J for

n ¼ 1; 2;y;(iii) Sn is uniformly distributed in J with density wd=ðaþdÞ as n-N:

Analogous results hold under the assumption that S;SnCJ:

A proof which is similar to that of Theorem 1, but with suitably distorted sets Sn

and which makes use of a result of Ewald et al. [14], yields the next result.

Theorem 2. Let f : ½0;þNÞ-½0;þNÞ satisfy the growth condition (2.1),

let a40 be the corresponding constant, and let jj � jj be a norm on Ed : Then

there is a constant div40; depending only on f and jj � jj; such that the following

holds: let JCEd be compact and measurable with jJj40; and w : J-Rþ continuous.

Then

(i) infSCEd

#S¼n

ZJ

minsAS

f f ðjjs ujjÞgwðuÞ du

� �Bdiv

ZJ

wðuÞd

aþd du

� �aþdd

f1

n1d

!as n-N:

If, in addition, J is connected and has positive density, see (2.2), then the following

assertions hold: let ðSnÞ be a sequence of sets SnCEd with #Sn ¼ n such that the

infimum in (i) is attained for S ¼ Sn; n ¼ 1; 2;y: Then

(ii) there is a constant d41 such that Sn is ð1=dn1=d ; d=n1=dÞ-Delone in J for n ¼1; 2;y;

(iii) Sn is uniformly distributed in J with ensity wd=ðaþdÞ as n-N:

Analogous results hold under the assumption that S;SnCJ:


Remark. Theorem 2 can be generalized to Finsler spaces where all tangent norms areisometric—as in Theorem 1. For general Finsler spaces there is also such a result, butthen the constant div depends on J:

The proof of Theorem 1 will be given in Sections 2.3–2.6. To show the Deloneproperty of Sn is the core of the proof. This property is then used to prove theasymptotic formula. The latter is the main tool for the proof that Sn is uniformlydistributed. We are aware that there are more direct proofs of the asymptoticformula; compare the sketch of the proof of the asymptotic formula in Theorem 2 in[33], or the proof of a similar formula due to Chernaya [8].

2.3. Preparations for the proof

First, let M be a Riemannian d-manifold. Let U be an open neighborhood of a

point pAM with corresponding homeomorphism h ¼ ‘‘0’’ which maps U onto the

open set U 0 ¼ hðUÞCEd : To each uAU 0 there corresponds a positive definite

quadratic form qu on Ed :

A curve C in U is of class C1 if it has a parametrization x : ½a; b -U such that

u ¼ h3x is a parametrization of class C1 of a curve in U 0: The length of C then isdefined to be

Z b

a

quðtÞð ’uðtÞÞ12 dt: ð2:3Þ

By means of appropriate dissection and addition one can define the length for any

curve in M which is piecewise of class C1 and which is not contained in a singleneighborhood. The Riemannian distance RMðx; yÞ of two points x; yAM is theinfimum of the lengths of continuous curves in M which connect x; y and are

piecewise of class C1: A curve in M of class C1 which connects two points x; yAM is

a (geodesic) segment if its length is RMðx; yÞ: A geodesic is a curve of class C1 in M

which consists locally of geodesic segments. A set NCM is geodesically convex if forany x; yAN there is a unique geodesic segment connecting x; y and this segment iscontained in N:

A set JCU is (Riemann or Jordan) measurable in M if hðJÞ is so in Ed : Then its(area) measure oMðJÞ is defined by

oMðJÞ ¼Z

hðJÞðdet quÞ

12 du;

where du ¼ du1?dud : Again, by dissection and addition one can define measur-ability and measure for sets which are not contained in a single neighborhood. (Foran alternative definition see 2.1.)


If JCU is measurable and w : J-R continuous, then the (Riemann) integral of w

on J is defined by ZJ

wðxÞ doMðxÞ ¼Z

hðJÞwðh1ðuÞÞðdet quÞ

12 du: ð2:4Þ

Also here one can define integrals on sets not contained in a single neighborhood.Using the exponential map (compare the proof of (2.12)) and simple arguments

involving M one can show the following:

Let ICM be compact. Then there are e41; W40 such that BMðp; RÞ is measurableand

Rd

epoMðBMðp; RÞÞpeRd for pAI ; 0oRpW: ð2:5Þ

Let J;w : J-Rþ be as in Theorem 1 and let l41: For each pAM choose U ; h ¼‘‘0’’; U 0 ¼ hðUÞ; qu; q ¼ qp0 as above where U is so small that the following claims

hold true:

1

lqðx0 y0Þ

12pRMðx; yÞplqðx0 y0Þ for x; yAU ;

1

lðdet qÞ

12pðdet quÞ

12plðdet qÞ

12 for uAU ;

1

ljK 0jðdet qÞ

12poMðKÞpljK 0jðdet qÞ

12 for measurable KCU ;

l inffwðxÞ : xAJ-UgXsupfwðxÞ : xAJ-Ug:

Let V be an open measurable neighborhood of p with cl VCU :As p ranges over the compact set J; the corresponding neighborhoods V form an

open covering of J: Thus there is a finite subcover. Therefore there are, say, m points

in J and corresponding neighborhoods Ul ;Vl ; homeomorphisms hl ¼ ‘‘0’’; andpositive definite quadratic forms ql ; such that

1

lqlðx0 y0Þ

12pRMðx; yÞpl qlðx0 y0Þ

12 for x; yAUl ; ð2:6Þ

1

lðdet qlÞ

12pðdet quÞ

12pl ðdet qlÞ

12 for uAU 0

l ; ð2:7Þ

1

ljK 0jðdet qlÞ

12poMðKÞpljK 0jðdet qlÞ

12 for measurable KCU ; ð2:8Þ

for l ¼ 1; 2;y;m: Moreover, the sets

Jl ¼ J-ðVl\ðV1,y,Vl1ÞÞCVlCUl ; l ¼ 1;y;m;


have the following properties:

J is the disjoint union of the measurable sets JlCVlCUl ;

where Vl is open and measurable; Ul open and cl VlCUl ; l ¼ 1;y;m: ð2:9Þ

wlpwðxÞplwl ;1l WlpwðxÞpWl for xAJl ;

where wl ¼ inffwðxÞ : xAJlg; Wl ¼ supfwðxÞ : xAJlg; l ¼ 1;y;m: ð2:10Þ

Since J is compact and w : J-Rþ continuous,

0ov ¼ inffwðxÞ : xAJgpsupfwðxÞ : xAJg ¼ WoN: ð2:11Þ

The following result will be used to show that Dirichlet–Voronoi cells aremeasurable; here int stands for interior.

For each point of M there is a geodesically convex compact neighborhood

N such that for any p; qAint N; paq; the bisector or Leibnizian plane

LNðp; qÞ ¼ fxAN : RMðp;xÞ ¼ RMðx; qÞg of p; q in N is measurable

with measure 0: ð2:12Þ

Since M is of class C3 with metric tensorfield of class C1; each point of M has acompact neighborhood N with the following properties: (i) NCV ; where V is anopen measurable neighborhood; (ii) N is geodesically convex; (iii) for each pAN thegeodesic segments connecting p with bd N cover N\fpg schlicht; (iv) for each

pAint N there is a diffeomorphism E1p : V-Ed ; the inverse of the exponential

mapping Ep; such that for each geodesic segment G connecting p with a point of bd N

the image E1p ðGÞ is a line segment starting at o: See [5] or [36].

Now, given a point of M; let N be the neighborhood of it just described.N is compact and it is geodesically convex by (ii). Next, let p; qAint N; paq:Then (ii) implies that each geodesic segment connecting p and bd N contains at most

one point of LNðp; qÞ: Clearly, LNðp; qÞ is closed in the compact set N: Hence

E1p ðLNðp; qÞÞ is closed in the compact set E1

p ðNÞ and each ray starting at o meets

E1p ðLNðp; qÞÞ at most once by (iii) and (iv). Thus E1

p ðLNðp; qÞÞ has Lebesgue

measure 0: Since it is compact it is (Jordan) measurable with (Jordan) measure 0:This yields (2.12).

Secondly, f will be investigated. Clearly, (2.1) implies the following:

Given 0oso1; the quotient f ðstÞ=f ðtÞ is increasing and has a positive

lower bound for t40: ð2:13Þ


As a consequence of (2.1) and (2.13) it will be shown that

There is a constant a40; depending on f ; such that the limit lðsÞ ¼lim

t-þ0ð f ðstÞ=f ðtÞÞ exists and is equal to sa for s40: ð2:14Þ

The existence and positivity of this limit follows from (2.1) and (2.13). Then

lðrsÞ ¼ limt-þ0

f ðrstÞf ðstÞf ðstÞf ðtÞ ¼ lðrÞlðsÞ for r; s40:

The function c : R-R defined by cðuÞ ¼ log lðeuÞ for uAR thus satisfies Cauchy’sfunctional equation. Since f is strictly increasing, (2.1) and (2.13) imply that lðsÞ41for s41 and lðsÞo1 for 0oso1; respectively. Hence cðuÞo0 for uo0 and cðuÞ40for u40: Being a solution of Cauchy’s functional equation it thus follows thatcðuÞ ¼ au for uAR; with suitable a40: See [37]. This yields (2.14).

The next required property of f is the following:

Define g : Rþ-Rþ by gðtÞ ¼ f ð1=t1=dÞ for t40: Then; given 0oZo1;

W; i40; Wai; there is a constant k40 such that

ZgðWtÞ þ ð1 ZÞgðitÞXð1 þ kÞgððZWþ ð1 ZÞiÞtÞ for sufficiently large t40:

ð2:15Þ

To see this note that by (2.14) and the definition of g hold:

gðWtÞB 1

Wad

gðtÞ; gðitÞB1

iad

gðtÞ;

gððZWþ ð1 ZÞiÞtÞB 1

ðZWþ ð1 ZÞiÞad

gðtÞ as t-N:

2.4. The Delone property of Sn; %Sn

Let M; J;w : J-Rþ be as in Theorem 1. Later on we will assume that J isconnected and has positive density.

At first tilings of J by Dirichlet–Voronoi cells will be investigated. Given S ¼fp1;y; pngCM; define the (Dirichlet–Voronoi) cell Di ¼ DðJ;S; piÞ of pi with respectto S in J by

Di ¼ fxAJ: RMðpi; xÞpRMðx; pjÞ for j ¼ 1;y; ng; i ¼ 1;y; n:

The circumradius Ri ¼ RðDiÞ of Di with respect to pi is given by

Ri ¼ inffR40 : BMðpi;RÞ*Dig; i ¼ 1;y; n:


With these definitions in mind the following will be shown:

Let S ¼ fp1;y; pngCM: If max1¼1;y;n

fRig is sufficiently small; then the

cells D1;y;Dn are measurable and form a tiling of J; ð2:16Þ

i.e., they cover J and their intersections have measure 0: There are finitely manyneighborhoods as described in (2.12) such that their interiors cover the compact setJ: Thus there is a constant 2m40; a Lebesgue number of this covering, see e.g. [13],with the following property: for each qAJ the ball BMðq; 2mÞ is contained in one ofthese neighborhoods. Assume now that maxi¼1;y;nfRigom: Then the following

holds:

oMðDi-DjÞ ¼ 0 for i; j ¼ 1;y; n; iaj: ð2:17Þ

If Di-Dj ¼ |; we are finished. Otherwise, let qADi-DjCJ: Then RMðpi; qÞ ¼RMðq; pjÞpRjom: Since Ri;Rjom; we see that Di-DjCBMðq; 2mÞ: Hence

Di-DjðCBMðq; 2mÞÞCN; where N is one of the neighborhoods as described in

(2.12). Since Di-DjCfxAN : RMðpi; xÞ ¼ RMðx; qjÞg it follows that Di-DjCLNðpi; pjÞ and thus oMðDi-DjÞpoMðLNðpi; pjÞÞ ¼ 0; by (2.12). The proof of

(2.17) is complete. Since D1;y;Dn; J are closed and D1,?,Dn ¼ J;

bd DiC[iaj;

Di-Dj

!,bd J for i ¼ 1;y; n:

This together with (2.17), the measurability of J which implies that oðbd JÞ ¼ 0; andthe fact that D1;y;Dn cover J yields the measurability of the cells D1;y;Dn andthe claim that they tile J; concluding the proof of (2.16).

The next preliminary step is to prove that

there is a constant n40 such that Fnpn f1

n1d

!for n ¼ 1; 2;y; ð2:18Þ

where for Fn see (i) in Theorem 1. Let l ¼ 2 and choose m and Ul ;y; l ¼ 1;y;m; as

in 2.3 such that (2.6)–(2.11) hold. Let nXm: Since the sets J 0l all are measurable in Ed

by (2.9), we may choose In=mm points in each J 0l such that the Euclidean balls with

centers at these points and radius Oð1=In=mm1dÞ cover J 0

l : Here Oð�Þ is a suitable

Landau symbol, the same for each l: If n is sufficiently large, which we assume, the

balls covering J 0l all are contained in U 0

l by (2.9). Consider the h1l -images of the

centers of these balls and let S be the union of all these images, l ¼ 1;y;m: Then#Spn and (2.6), (2.7), (2.9), (2.11), the definition of integrals on M; and (2.14)together yield (2.18).

Assume from now on in this subsection that J is connected and has positivedensity, i.e., satisfies (2.2), but note that these assumptions will only be needed


beginning with the proof of proposition (2.26). Let Sn ¼ fpn1;y; pnng and Dni ¼DðJ;Sn; pniÞ;Rni ¼ RðDniÞ for i ¼ 1;y; n; n ¼ 1; 2;y : Our aim is to show thefollowing:

There is a constant d41 such that

1

dn1d

p mini; j¼1;y;n

iaj

fRMðpni; pnjÞg; maxi¼1;y;n

fRnigpd

n1d

for n ¼ 1; 2;y : ð2:19Þ

We show the upper estimate first. Its proof is split into a series of steps. In the firststep it will be shown that

Dnia| for i ¼ 1;y; n; n ¼ 1; 2;y : ð2:20Þ

If Dni ¼ |; then

minpASn

fRMðp; xÞg ¼ minpASn\fpnig

fRMðp; xÞg for xAJ:

Now choose qAðint JÞ\Sn and let Tn ¼ ðSn\fpnigÞ,fqg: Then

minpASn

fRMðp; xÞgXminpATn

fRMðp; xÞg for xAJ;

where strict inequality holds for all xAJ sufficiently close to q: Since f is strictlyincreasing by (2.1), S ¼ Sn does not minimize the left hand integral in (2.3). Thiscontradicts our choice of Sn and thus concludes the proof of (2.20).

The next step is to show that

maxxAJ

minpASn

fRMðp; xÞg� �

-0 as n-N: ð2:21Þ

For suppose not. Then there are x40 and a sequence of points xnAJ where n

is from an infinite subsequence of 1; 2;y; such that RMðp; xnÞX2x for eachpASn: Since J is compact, there is a point xAJ such that RMðp; xÞXx for each pASn

and n from a suitable infinite subsequence of the subsequence just considered.Combining this with the assumptions that J has positive density (see (2.2)) and f isstrictly increasing (see (2.1)), we obtain a contradiction to (2.18). This concludes theproof of (2.21).

(2.20), (2.21) and (2.16) yield the following:

maxi¼1;y;n

fRnig-0 as n-N; ð2:22Þ

the cells Dn1;y;Dnn are measurable and form a tiling of J for all

sufficiently large n; ð2:23Þ

there is a compact set I*J such that SnCI for n ¼ 1; 2;y : ð2:24Þ


The definition of Fn in (i), our choice of Sn ¼ fpn1;y; pnng and of Dn1;y;Dnn;(2.23), the assumptions that f is increasing and wX0 together imply the nextassertion:

Let Tn ¼ fqn1;y; qnngCM and let En1;y;EnnCJ be measurable and form a tilingof J for n ¼ 1; 2;y : Then, if n is sufficiently large,

Fn ¼Z

J

mini¼1;y;n

f f ðRMðpni; xÞÞgwðxÞ doMðxÞ

¼Xn

i¼1

ZDni

f ðRMðpni; xÞÞwðxÞ doMðxÞ

pZ

J

mini¼1;y;n

f f ðRMðqni; xÞÞgwðxÞ doMðxÞ

pXn

i¼1

ZEni

f ðRMðqni; xÞÞwðxÞ doMðxÞ: ð2:25Þ

Before proving the upper estimate in (2.19), a weaker version of it will be shown inthe following step:

There is a constant s40 such that Rnjps

n1d

for suitable jAf1;y; ng;

and all n ¼ 1; 2;y : ð2:26Þ

If (2.26) did not hold, then,

for any ðarbitrarily largeÞ z40 there is an infinite set of n such that

mini¼1;y;n

fRnigXz

n1d

: ð2:27Þ

Take b from (2.2) and v;W from (2.11). By (2.14) one may

choose z40 so large that for suitable t40 holds

oMðJÞWf ð3tÞozdv

3dbf

2t

3

� � f

t

3

� �� for 0otpt: ð2:28Þ

From now on we consider in the proof of (2.26) only such n for which

ð2:23Þ; ð2:25Þ; and ð2:27Þ for the chosen z hold; and ð2:29Þ

maxi¼1;y;n

fRnigpg; t: ð2:30Þ


Here for (2.30) we have made use of (2.22) and g; t are as in (2.2) and (2.28),respectively. By (2.29) and (2.23) the cells Dn1;y;Dnn are measurable and form atiling of J: Hence there is a cell Dnj ; say, such that

oMðDnjÞpoMðJÞ

n: ð2:31Þ

Since in this subsection J is assumed to be connected and since the cells Dn1;y;Dnn

tile J by (2.29) and (2.23) and are compact, we may choose a point pADnj-Dnk for

suitable kaj: Then RMðpnj ; pnkÞpRMðpnj ; pÞ þ RMðp; pnkÞ ¼ 2RMðpnj ; pÞp2Rnj and

therefore,

kaj and RMðpnk; xÞðpRMðpnk; pnjÞ þ RMðpnj ; xÞÞp3Rnj for xADnj: ð2:32Þ

Choose qADnj with RMðpnj ; qÞ ¼ Rnj : The definition of Dnj then shows that

RMðq; xÞpRnj

3; RMðpni; xÞX2Rnj

3for xAB ¼ BM q;

Rnj

3

� �; i ¼ 1;y; n: ð2:33Þ

Define fn : J-R by

fnðxÞ ¼ mini¼1;y;n

f f ðRMðpni; xÞÞg for xAJ: ð2:34Þ

Finally, the definition of Fn in (i), our choice of Sn ¼ fpn1;y; pnng; (2.34), (2.29) and(2.23), the measurability of B which follows from (2.30) and (2.2), (2.25), theassumptions that f is increasing and f ;wX0; (2.32),(2.34), (2.11), (2.30), (2.2), (2.34)and (2.33), (2.11), (2.3),(2.34), (2.31), (2.29) and (2.27), (2.30), and (2.28) lead to thefollowing contradiction and thus conclude the proof of (2.26):

Fn ¼Z

J

fnðxÞwðxÞ doMðxÞ ¼Xn

i¼1

ZDni


pXn

i¼1iaj

ZDni\B

f ðRMðpni; xÞÞwðxÞ doMðxÞ þZ

Dnj \B

f ðRMðpnk; xÞÞwðxÞ doMðxÞ

þZ

B-J

f ðRMðp; xÞÞwðxÞ doMðxÞ


pXn

i¼1iaj

ZDni\B


Dnj \B

f ðRMðpnj ; xÞÞwðxÞ doMðxÞ

þZ

Dnj \B

f ð3RnjÞwðxÞ doMðxÞ þZ

B-J

fnðxÞwðxÞ doMðxÞ

Z

B-J

ð fnðxÞ f ðRMðq; xÞÞÞwðxÞ doMðxÞ

pZ

J

fnðxÞwðxÞ doMðxÞ þ oMðDnjÞ f ð3RnjÞW

Rd

nj

3dbf

2Rnj

3

� � f

Rnj

3

� �� v

pFn þ1

noMðJÞW f ð3RnjÞ

1

n

zdv

3dbf

2Rnj

3

� � f

Rnj

3

� �� oFn:

We come to the final step of the proof of the upper estimate in (2.19). If thisestimate did not hold, then

for any (arbitrarily large) n40 there is an infinite set of n such that

Rnk4n

n1d

for suitable kAf1;y; ng: ð2:35Þ

Take b from (2.2), e from (2.5), where I is from (2.24), s from (2.26), and v;W from(2.11). By (2.14) we may choose n40 so large that

(i) n4s;(ii) for suitable j40 hold

Wesdf3s

n1d

! vnd

3dbf

2t

3

� � f

t

3

� �� o0 for

n

n1d

otoj: ð2:36Þ

From now on consider in the proof of the upper estimate in (2.19) only such n that

ð2:5Þ; where I is from ð2:24Þ; ð2:23Þ; ð2:25Þ and ð2:35Þ hold; and ð2:37Þ

maxi¼1;y;n

fRnigpg; W;j; ð2:38Þ

where g; W;j are as in (2.2), (2.5) and (2.36), respectively. Given such an n; let j be as

in (2.26). Then Rnjps=n1=d : Choose pnlapnj with RMðpnj ; pnlÞp2Rnjp2s=n1=d : Such

a choice is possible since in this subsection J is connected by assumption, compare


the argument that led to (2.32). Then

laj and RMðpnl ; xÞp3Rnjp3s

n1d

for xADnj : ð2:39Þ

By (2.37), proposition (2.35) holds for n and the chosen n: Let k be as in (2.35). Since

Rnjps=n1=don=n1=doRnk by (2.26), (2.36i) and (2.37) and (2.35),

kaj ð2:40Þ

follows. Choose pADnk with RMðpnk; pÞ ¼ Rnk: The definition of the cell Dnk

and (2.35) then imply that RMðpni; pÞXRMðpnk; pÞ ¼ Rnk4n=n1=d for i ¼ 1;y; n:Hence,

RMðp; xÞpRnk

3; RMðpni; xÞX2Rnk

3for xAB ¼ BM p;

Rnk

3

� �; i ¼ 1;y; n: ð2:41Þ

Finally, the definition of Fn in (i), our choice of Sn ¼ fpn1;y; pnng; (2.37) and (2.25),the measurability of B which follows from (2.38) and (2.2), (2.39), (2.40), (2.37),(2.23) and (2.25), the assumptions that f is increasing and f ;wX0; (2.39), (2.34),(2.11), (2.41), (2.38), (2.2), (2.11), (2.38), (2.26), (2.37) and (2.5), (2.26), (2.37), (2.35),(2.38) and (2.36) together yield a contradiction and thus conclude the proof of theupper estimate in (2.19):

Fn ¼Xn

i¼1

ZDni


pXn

i¼1iaj;k

ZDni\B


Dnj \B

f ðRMðpnl ; xÞÞwðxÞ doMðxÞ

þZ

Dnk\B

f ðRMðpnk; xÞÞwðxÞ doMðxÞ þZ

B-J


pXn

i¼1iaj;k

ZDni\B


Dnj \B


þZ

Dnj \B

f3s

n1d

!wðxÞ doMðxÞ þ

ZDnk\B


þZ

B-J

fnðxÞwðxÞ doMðxÞ Z

B-J

ð fnðxÞ f ðRMðp; xÞÞÞwðxÞ doMðxÞ

¼Z

J

fnðxÞwðxÞ doMðxÞ þ oMðDnjÞf3s

n1d

!W vRd

nk

3dbf

2Rnk

3

� � f

Rnk

3

� ��

pFn þ1

nWe sdf

3s

n1d

! 1

n

vnd

3dbf

2Rnk

3

� � f

Rnk

3

� �� oFn:


We proceed to the proof of the lower estimate in (2.19). The proof will be split intotwo steps. In the first step it will be shown that

there is a constant f40 such that for j ¼ 1;y; n; holds

FnpZ

J

gn1ðxÞwðxÞ doMðxÞ fn

f1

n1d

!; gn1ðxÞ ¼ min

i¼1;y;niaj

f f ðRMðpni; xÞÞg: ð2:42Þ

It is enough to show this for sufficiently large n: The argument that led to (2.32)shows that for each xAJ there is a point in fpn1;y; pnj1; pnjþ1;y; pnng at distance

at most 3d=n1d ; where d is from the upper estimate in (2.19). Hence the cells in J

corresponding to the set fpn1;y; pnj1; pnjþ1;y; pnng have circumradius at most

3d=n1=d : Then (2.16) shows that for all sufficiently large n and any j ¼ 1;y; n; then 1 cells corresponding to fpn1;y; pnj1; pnjþ1;y; pnng are measurable and tile J:

Thus there is a cell of measure at least oMðJÞ=ðn 1Þ4oMðJÞ=n; say the cellcorresponding to pnk: Clearly, kaj: Noting (2.24) and (2.5) we see that this cell has

circumradius at least w=n1=d ; where w40 is a suitable constant. Hence there is a point

p of this cell with RMðpnk; pÞXw=n1=d : The definition of cells then shows that

RMðpni; pÞXw=n1=d for i ¼ 1;y; d; iaj: As before,

RMðp; xÞp w

3n1d

; RMðpni; xÞX 2w

3n1d

for xAB ¼ BM p;w

3n1d

!; i ¼ 1;y; n; iaj; ð2:43Þ

compare (2.33) and (2.41). Now, the definitions of Fn in (i) and of gn1 in (2.42), themeasurability of B which follows from (2.2) for sufficiently large n; the assumptionthat J satisfies (2.2), (2.43), the assumptions that f is increasing and f ;wX0; and(2.11) imply, for sufficiently large n; that

FnpZ

J

min mini¼1;y;n

iaj

f f ðRMðpni; xÞÞg; f ðRMðp; xÞÞ

8<:

9=;wðxÞ doMðxÞ

¼Z

J

minfgn1ðxÞ; f ðRMðp;xÞÞgwðxÞ doMðxÞ

pZ

J\B

gn1ðxÞwðxÞ doMðxÞ þZ

B-J


¼Z

J

gn1ðxÞwðxÞ doMðxÞ Z

B-J

ðgn1ðxÞ f ðRMðp;xÞÞÞwðxÞ doMðxÞ

pZ

J

gn1ðxÞwðxÞ doMðxÞ 1

n

wdv

3dbf

2w

3n1d

! f

w

3n1d

! !:

An application of (2.14) then yields (2.42).


In the second step we assume that the lower estimate in (2.19) does not hold. Then,

for any ðarbitrarily smallÞ c40 there is an infinite set of n such that

RMðpnj; pnkÞpc2

n1d

for suitable j; kAf1;y; ng; jak: ð2:44Þ

Let d; e;f;W be as in the upper estimate in (2.19), (2.5), where I is from (2.24),(2.42), and (2.11), respectively. By (2.14) we may choose 0oo;co1 so small that

bddoWfd

n1d

!þ bcdWf

2c

n1d

!off

1

n1d

!; ð2:45Þ

f ðð1 þ cÞtÞpð1 þ oÞf ðtÞ for 0otpt; ð2:46Þwith a suitable t40: In the remaining part of the proof of the lower estimate in (2.19)we consider only n for which

ð2:25Þ and ð2:44Þ hold; and ð2:47Þ

maxi¼1;y;n

fRnigpd

n1d

pminfg; tg; c

n1d

pg; ð2:48Þ

where g is as in (2.2) and t as in (2.46). For (2.48) we have used the upper estimate in(2.19). Propositions (2.47) and (2.44) and our choice of cðo1Þ imply that

RMðpnk; xÞpð1 þ cÞRMðpnj ; xÞ for xeB ¼ BM pnj;c

n1d

!;

RMðpnk; xÞp2c

n1d

for xAB: ð2:49Þ

Now, combining (2.42), (2.47) and (2.25) applied with n 1 instead of n; whereEni ¼ Dni for i ¼ 1;y; n; iaj; k and Enk ¼ Dnj,Dnk; the measurability of B which

follows from (2.24) and (2.5) for sufficiently large n; the assumptions that f isincreasing and f ;wX0; (2.49), (2.46), (2.48) and (2.2), (2.11), (2.48) and (2.2), theassumptions that f is increasing and f ;wX0 and (2.45), we obtain for sufficientlylarge n the following contradiction which concludes the proof of the lower estimatein (2.19):

FnpZ

J

mini¼1;y;n

iaj

f f ðRMðpni; xÞÞgwðxÞ doMðxÞ fn

f1

n1d

!

pXn

i¼1iaj

ZDni


Dnj


fn

f1

n1d

!


pXn

i¼1iaj

ZDni


Dnj \B

f ðð1 þ cÞRMðpnj ; xÞÞwðxÞ doMðxÞ

þZ

B-Dnj

f2c

n1d

!wðxÞ doMðxÞ f

nf

1

n1d

!

pXn

i¼1iaj

ZDni


Dnj \B


þ oZ

Dnj \B

f ðRMðpnj ; xÞÞwðxÞ doMðxÞ þ 1

nbcdW f

2c

n1d

! f

nf

1

n1d

!

pFn þ1

nb ddoW f

d

n1d

!þ 1

nb cdW f

2c

n1d

! f

nf

1

n1d

!oFn:

For n ¼ 1; 2;y; choose %Sn ¼ f %pn1;;y; %pnngCJ in the following way: if in (i) the

infimum is considered only for S with #S ¼ n for which SCJ (instead of SCM),

then it is attained for S ¼ %Sn: If J is connected and has positive density, then a proofsimilar to the one which led to (2.19), but easier in some details, shows that

there is a constant %d41 such that

1

%dn1d

p mini;j¼1;y;n

iaj

fRMð %pni; %pnjÞg; maxi¼1;y;n

f %Rnigp%d

n1d

for n ¼ 1; 2;y; ð2:50Þ

where the %Rni are the circumradii of the cells in J corresponding to f %pn1;y; %pnng:

2.5. The constant div

Let M ¼ Ed ; J ¼ ½0; 1 d ;w ¼ 1: If the infimum in (i) is attained for, say, Tn ¼ftn1;y; tnng; then TnC½0; 1 d : Thus an application of (2.19) or (2.50), and (2.14)together with the assumptions that f is increasing and fX0 yields the following: if

Gn ¼ infTCEd

#T¼n

Z½0;1 d

mintAT

f f ðjjt ujj2Þg du

( )¼Z½0;1 d

mini¼1;y;n

f f ðjjtni ujj2Þg du;

then Gn=f ð1=n1dÞ is bounded between positive constants. Thus, defining

div ¼ lim infn-N

Gn

f1

n1d

!; ð2:51Þ


it follows that

divARþ: ð2:52Þ

We will show that

div ¼ limn-N

Gn

f1

n1d

!: ð2:53Þ

Let d41 be as in (2.19). Choose m41 (arbitrarily close to 1) and let m be such that

Gm

f1

m1d

!pm div;

fd

km1d

!

f1

km1d

!pm

fd

m1d

!

f1

m1d

! for k ¼ 1; 2;y: ð2:54Þ

This is possible by the definition of div in (2.51), (2.52) and (2.1). An application of(2.13) shows that

fd

km1d

!

fd

m1d

! f ðtÞXft

k

� �for 0otp

d

m1d

; k ¼ 1; 2;y : ð2:55Þ

For k ¼ 1; 2;y; dissect the cube ½0; 1 d in the usual way into kd cubelets, each of

edge length 1=k: The kd affine transformations

Aa : u1-u1 þ a1

k;y; ud-

ud þ ad

k; z-

fd

km1d

!

fd

m1d

! z for ðu; zÞAEdþ1;

where a ¼ ða1;y; adÞTAf0;y; k 1gd ;

map the set

fðu; zÞAEdþ1 : uA½0; 1 d ; 0pzp mini¼1;y;m

f f ðjjtmi ujj2Þgg:

onto kd non-overlapping sets. Together, these sets contain the set

fðu; zÞAEdþ1 : uA½0; 1 dg; 0pzp mini¼1;y;m

aAf0;y;k1gd

f f ðjjAatmi ujj2Þg:


This follows from (2.55) and the fact that by (2.19) the cells in ½0; 1 d corresponding

to Tm ¼ ftm1;y; tmmg have circumradius at most d=m1d : Hence the definition of Gkd m

and (2.54) show that

Gkd mpkd � 1

kd

fd

km1d

!

fd

m1d

! Gmpm

f1

km1d

!

f1

m1d

! Gm;

and thus, again by (2.54),

Gkd m

f1

km1d

!pmGm

f1

m1d

!pm2 div:

By (2.14) there is k0 such that

f1

km1d

!

f1

ðk þ 1Þm1d

0@

1Apm for kXk0:

Finally, given nXkd0 m; let kXk0 be such that kdmpnoðk þ 1Þd

m: Thus,

Gn

f1

n1d

! pGkd m

f1

km1d

!f

1

km1d

!

f1

ðk þ 1Þm1d

0@

1A

0BBBBBB@

1CCCCCCApm3 div for all sufficiently large n:

Since m41 was arbitrary, this together with (2.51) implies (2.53).Next, (2.53) will be extended to arbitrary cubes:

Let C be a cube in Ed : Then

Hm ¼ infTCEd

#T¼m

ZC

mintAT

f f ðjjt ujj2Þg du

� �BdivjCj

aþdd f

1

m1d

!as m-N; ð2:56Þ

where a is as in (2.14). We may assume that C ¼ n½0; 1 d ¼ ½0; n d for suitable n40:

By (2.19) the circumradii of the cells corresponding to TmC½0; 1 d are bounded above


by d=m1=d : Thus by (2.14),

Gm ¼Z½0;1 d

mintATm

f f ðjjt ujj2Þg duBZ½0;1 d

minntAnTm

f f ðjjnt nujj2Þg du1

na

¼Z½0;n d

minvAnTm

f f ðjjv wjj2Þg dw1

naþdX

Hm

naþd¼ Hm

jCjaþd

d

;

or

GmjCjaþd

d \Hm as m-N:

The same argument with ½0; 1 d and C exchanged shows that

Hm\GmjCjaþd

d as m-N;

and therefore,

HmBGmjCjaþd

d as m-N:

This together with (2.53) yields (2.56).A suitable affine transformation applied to (2.56) finally gives the following:

Let qð�Þ be a positive definite quadratic form on Ed and C0 a cube in

Ed with respect to the norm qð�Þ1=2: Then

infT 0CEd

#T 0¼m

ZC0

mintAT 0

f f ðqðt uÞ12Þg du

� �Bdiv ðdet qÞ

a2d jC0j

aþdd f

1

m1d

!as m-N: ð2:57Þ

2.6. The asymptotic formula

Choose Sn ¼ fpn1;y; pnng and Dn1;y;Dnn for n ¼ 1; 2;y; as in 2.4. Define

SnðKÞ ¼ fqniASn : Dni-Ka|g; nðKÞ ¼ #SnðKÞ for measurable KCJ:

If in the following an integral is written omitting the integrand, the integrand is to be

wðxÞd=ðaþdÞ and byR a

we mean the ath power ofR:

In the first part of the proof of the asymptotic formula (i) it will be shown that

Fn\div

ZJ

wðxÞd


d

f1

n1d

!as n-N: ð2:58Þ


Let l41 and choose Ul ;Vl ;y; for l ¼ 1;y;m; as in 2.3 such that

ð2:6Þ2ð2:11Þ hold: ð2:59Þ

The first step of the proof of (2.58) is to show that

there is a constant n41 such that1

nf

1

n1d

!pFnpn f

1

n1d

!

for n ¼ 1; 2;y ð2:60Þ

The upper estimate was proved in (2.18). For the proof of the lower estimate choose

a set C in Jl ; say, such that C0 is a cube with respect to the norm qlð�Þ12: Then C is

measurable and the definition of Fn in (i), (2.11), the assumptions that f is increasingand f ;wX0; (2.59) and (2.6), the definition of integrals in M; (2.59) and (2.7)show that

FnX

ZC

minpASn

f f ðRMðp; xÞÞgwðxÞ doMðxÞ

X1

lvðdet qlÞ

12

ZC0

minsASnðCÞ0

f1

lqlðs uÞ

12

� �� du

X1

lv ðdet qlÞ

12 inf

T 0CEd

#T 0¼nðCÞ

ZC0

mintAT 0

f1

lqlðt uÞ

12

� �� du

� �:

If nðCÞ did not tend to N as n-N; the latter expression would not tend to 0; incontradiction to the upper estimate for Fn in (2.60). Hence nðCÞ-N as n-N:Thus, using (2.14), (2.57), and the assumption that f is increasing, we concludefurther that

Fn\1

l1þa v ðdet qlÞaþd2d divjC0j

aþdd f

1

nðCÞ1d

0@

1A

X1

l1þa v ðdet qlÞaþd2d divjC0j

aþdd f

1

n1d

!as n-N:

This readily implies the lower estimate in (2.60), concluding the proof of (2.60).

Noting that Fnpnf ð1=n1=dÞ by (2.60), the assumption that f is strictly increasing,and (2.14), we obtain the following proposition as a byproduct of the last part of theproof of (2.60):

Let CCJl ; say; such that C0 is a cube with respect to the norm qlð�Þ12:

Then there is a constant x with 0oxo1 such that xnpnðCÞpn for

sufficiently large n: ð2:61Þ


We come to the second step of the proof of (2.58). Noticing (2.9), we may choosesets CliCJl ; i ¼ 1;y; il ; l ¼ 1;y;m; with the following properties:

The sets CliCJ; i ¼ 1;y; il ; l ¼ 1;y;m; are compact and pairwise

disjoint: ð2:62Þ

C0li is a cube with respect to the norm qlð�Þ

12: Thus; in particular;

Cli is measurable: ð2:63Þ

Xl;i

ZCli

doMðxÞX1

l

ZJ

doMðxÞ: ð2:64Þ

By the definition of SnðCliÞ; (2.62) and (2.22) (which has been proved without theassumptions that J is connected and has positive density),

the sets SnðCliÞ are pairwise disjoint for sufficiently large n: Thus; ð2:65Þ

Xl;i

nðCliÞpn for sufficiently large n: ð2:66Þ

Finally, it follows from (2.63) and (2.61) that

there is a constant s with 0oso1 such that snpnðCliÞpn

for i ¼ 1;y; il ; l ¼ 1;y;m; and sufficiently large n: ð2:67Þ

In the third step note that (2.60), (2.67) and (2.66) together yield the followingpropositions, where

L ¼ lim infn-N

Fn

f1

n1d

! ð40Þ:

There is an infinite subsequence of 1; 2;y; such that Fn=f ð1=n1dÞ-L

as n-N in this subsequence: ð2:68Þ

There are constants sli40 such that nðCliÞBslin

for i ¼ 1;y; il ; l ¼ 1;y;m; as n-N in the same subsequence as in ð2:68Þ:Furthermore;

Xl;i

slip1: ð2:69Þ


In the final step of the proof of (2.58) assume that n is from the subsequenceconsidered in (2.68) and (2.69) and that it is so large that (2.65) and (2.66)hold. Then the definition of Fn in (i), our choice of Sn; (2.62), (2.63), the definitionof SnðCliÞ; the definition of integrals on M; the assumptions that f is increasingand f ;wX0; (2.59) and (2.6), (2.10), the inclusion CliCJl ; (2.14), (2.57), (2.69),the definition of integrals on M; (2.59) and (2.6), (2.10), (2.14), (2.64),(2.15), the assumption that f is increasing, (2.64), (2.69), and (2.14) imply thefollowing:

Fn ¼Z

J

minpASn


X

Xl;i

ZCli

minpASnðCliÞ

f f ðRMðp;xÞÞgwðxÞ doMðxÞ

X1

l

Xl;i

ðdet qlÞ12

ZC0

li

minsASnðCliÞ0

f1

lqlðs uÞ

12

� �� wl du

B1

l1þa divX

l;i

ðdet qlÞaþd2d jC0

lijaþd

d wl f1

ðslinÞ1d

0@

1A

X1

l1þaþ1þaþdd

divX

l;i

ZCli

wðxÞd


d

f1

ðslinÞ1d

0@

1A

B1

l3þaþad

divX

l;i

ZCli

f

RCli

slin

!1d

0B@

1CA

¼ 1

l3þaþad

divX

l;i

ZCli

Xl;i

RCliP

l;i

RCli

f

1

slinRCli

!1d

0BB@

1CCA

\1

l4þaþad

div

ZJ

f

1

Pl;i

RCli

slinRCli

Pl;i

RCli

0B@

1CA

1d

0BBBBB@

1CCCCCA

¼ 1

l4þaþad

div

ZJ

f

Pl;i

RCliP

l;islin

!1d

0B@

1CA

X1

l4þaþad

div

ZJ

f1

l

ZJ

1

n

� �1d

0@

1AB

1

l4þaþ2ad

div

Z aþdd

J

f1

n1d

!


as n-N in the subsequence considered in (2.68) and (2.69). Since l41 wasarbitrary, we have thus shown that

lim infn-N

Fn

f1

n1d

!Xdiv

ZJ

wðxÞd


d

;

which is equivalent to (2.58).In the second part of the proof of the asymptotic formula (i) we consider, in

addition to Fn; the quantity

%Fn ¼ inf%SCJ

# %S¼n

ZJ

min%pA %S

f f ðRMð %p; xÞÞgwðxÞ doMðxÞ� �

:

Our aim is to show that

Fnp %Fntdiv

ZJ

wðxÞd


d

f1

n1d

!as n-N: ð2:70Þ

Let l41 and choose Ul ;Vl ;y; l ¼ 1;y;m; as in 2.3 such that

ð2:6Þ2ð2:11Þ hold: ð2:71Þ

In the first step of the proof of (2.70) we choose sets Cli; i ¼ 1;y; il ; l ¼ 1;y;m;with the following properties, where s40 (is small) and Wl is as in (2.10):

For each l ¼ 1;y;m; the sets Cli; i ¼ 1;y; il ; are compact;

non-overlapping and such that

JlCCl1,?,ClilCVl ;Cli-Jla|: ð2:72Þ

C0li is a cube with respect to the norm qlð�Þ

12 ðthus; in particular;

Cli is measurableÞ and such that

ðdet qlÞ12jC0

lijWd

aþdl ¼ s

ZJ

wðxÞd

aþd doMðxÞ: ð2:73Þ

Xl;i

ZCli

plZ

J

: ð2:74Þ

Let i0 ¼ i1 þyþ im: Since

Xl;i

ðdet qlÞ12jC0

lijWd

aþdl pl

daþd

þ1X

l;i

ZCli

wðxÞd

aþd doMðxÞpl3

ZJ

wðxÞd

aþd doMðxÞ


by (2.71) and (2.10), (2.7), the definition of integrals on M; and (2.74), it followsfrom (2.73) that

si0pl3: ð2:75Þ

By (2.14) we may choose k0 so large that

1

iad0

f1

k1d

!pl f

1

ði0kÞ1d

0@

1A for kXk0; ð2:76Þ

f1

ði0kÞ1d

0@

1Apl f

1

ði0ðk þ 1ÞÞ1d

0@

1A for kXk0: ð2:77Þ

In the next step of the proof of (2.70) we consider first the case where n is of theform n ¼ i0k for some kXk0: Assume that

infT 0CEd

#T 0¼k

ZC0

li

mintAT 0

f f ðqlðt uÞ12Þg du

( )

is attained for T 0 ¼ T 0lik where T 0

likCC0li;#T 0

lik ¼ k: ð2:78Þ

Applying the result (2.19) for M ¼ Ed and J ¼ C0li; i ¼ 1;y; il ; l ¼ 1;y;m; we

obtain the following, where, omitting indices, T 0lik ¼ ft1;y; tkg and D0

1;y;D0k are

the cells in C0li corresponding to the set T 0

lik and the norm qlð�Þ1=2:

There is a t41; independent of l; i; k; such that any two distinct points

from T 0lik ¼ ft1;y; tkg have distance at least 1=tk1=d and each of the

cells D01;y;D0

k has circumradius at most t=k1=d ðwith respect to the

norm qlð�Þ1=2Þ: ð2:79Þ

Since J is (Jordan) measurable, bd J is measurable and oMðbd JÞ ¼ 0: Hence

ðCli-bd JÞ0 is measurable in Ed with jðCli-bd JÞ0j ¼ 0: A result on Jordan

measurable sets of measure 0 in Ed then says that the parallel set[uAðCli-bd JÞ0

ðu þ nBdÞ

has arbitrarily small measure if n40 is sufficiently small. Here Bd is the solid

Euclidean unit ball in Ed : This remark together with (2.79) yields the next

proposition, where Dj ¼ h1l ðD0

jÞ:

There are at most oðkÞ cells D0j in C0

li with ðDj-bd JÞ0a|; where we

may choose the same Landau symbol oðkÞ for all l; i: ð2:80Þ


If D0jCðCli-int JÞ0; let %tj ¼ tj; if D0

j-ðCli-bd JÞ0a|; choose for %tj an arbitrary point

of D0j-ðCli-bd JÞ0: Let %T 0

lik be the set of all points %tj which can be obtained in this

way and let %Tlik ¼ h1l ð %T 0

likÞCCli: Then the definition of integrals on M in (2.4), the

assumptions that f is increasing and f ;wX0; (2.72), (2.71) and (2.6), (2.7), (2.10),(2.14), (2.2), (2.79),(2.80), (2.57), and (2.73) yield the following:Z

Cli-J

min%pA %Tlik

f f ðRMð %p; xÞÞgwðxÞ doMðxÞ

plZðCli-JÞ0

min%tA %T0

lik

f f ðlqlð%t uÞ12Þg du Wlðdet qlÞ

12

tl1þaZ

C0li

mintATlik

f f ðqlðt uÞ12Þg du Wlðdet qlÞ

12 þ oðkÞb 2dtd

kf

2t

k1d

!

Bl1þa div ðdet qlÞaþd2d jC0

lijaþd

d Wl f1

k1d

!þ oð1Þ f

1

k1d

!

tl1þaþ1þaþdd div s

ZJ

wðxÞd


d

f1

k1d

!as k-N;

where the symbols oðkÞ; oð1Þ are independent of l; i: ð2:81Þ

For n ¼ iok let %Tn ¼S

l;i%Tlik: Then # %Tnpn and %TnCJ: The definitions of Fn and %Fn;

(2.72), (2.71) and (2.9), i0 ¼ i1 þ?þ im; (2.81), (2.75) and (2.14) together imply that

Fnp %FnpZ

J

min%pA %Tn


pX

l;i

ZCli-J

min%pA %Tlik


tl3þ2ai0 div sZ

J

wðxÞd


d

f1

k1d

!

¼ l3þ2a div

ZJ

wðxÞd


d ði0sÞaþd

d

ia0df

1

k1d

!

tl3þ2aþ3aþ3dd div

ZJ

wðxÞd


d

f1

n1d

!as n ¼ iok-N:

Since l41 was arbitrary, this proves (2.70) for n of the form n ¼ iok: This, togetherwith (2.77) yields (2.70) for general n; compare the argument in the last part of theproof of (2.53).

Having proved (2.58) and (2.70), the proof of the asymptotic formula (i) with

SCM and also with %SCJ is complete.


2.7. Uniform distribution of Sn; %Sn

Assume that J is connected and has positive density, i.e. satisfies (2.2). We willprove that

Sn is uniformly distributed in J with respect to the density wd=ðaþdÞ

as n-N: ð2:82Þ

A proof which is similar to that of (2.80), but slightly more complicated in several

details since it deals with M instead of Ed ; and which makes use of (2.19), (2.24), and(2.5) yields the following:

Let ZCJ be measurable with oMðZÞ ¼ 0:

Then nðZÞð¼ #fi : Dni-Za|gÞ ¼ oðnÞ as n-N: ð2:83Þ

More difficult is the proof of

Let KCJ be measurable with 0ooMðKÞpoMðJÞ: Then

nðKÞBZðKÞn as n-N; where ZðKÞ ¼Z

K

ZJ

�: ð2:84Þ

If oMðKÞ ¼ oMðJÞ; then (2.84) follows from (2.20), which shows that nðJÞ ¼ n; andfrom (2.83). Suppose now that oMðKÞooMðJÞ and assume that (2.84) does not hold:

nðKÞfZðKÞn as n-N: ð2:85Þ

From (2.18) and the definitions of SnðKÞ; nðKÞ in 2.6 it follows that

n f1

n1d

!XFnX

ZK

minpASn


¼Z

K

minpASnðKÞ

f f ðRMðp;xÞÞgwðxÞ doMðxÞ

X infTCM

#T¼nðKÞ

ZK

minpAT

f f ðRMðp; xÞÞgwðxÞ doMðxÞ� �

:

This can hold only if nðKÞ-N as n-N: Now, applying the asymptotic formula (i)to K instead of J; we conclude further that

n f1

n1d

!\div

ZK

aþdd f

1

nðKÞ1d

0@

1A as n-N:

Using the assumption that f is strictly increasing and (2.14) it follows that nðKÞ=n isbounded below by a positive constant. An analogous statement holds for the set


L ¼ J\K : By (2.20) and (2.83),

nðKÞ þ nðLÞ ¼ n þ oðnÞ as n-N: ð2:86Þ

(2.85), (2.86) and the above statements on nðKÞ=n and nðLÞ=n show that

there are W; i40 such that Wo1oi or io1oW and nðKÞB WZðKÞn;

nðLÞBiZðLÞn as n-N; where n is from a suitable subsequence

of 1; 2;y : ð2:87Þ

From now on consider in the proof of (2.82) only n from the subsequence in (2.87). We

consider only the case that Wo1oi: By (i), the definition of Sn; J ¼ K,L;K-L ¼ |;the definitions of SnðKÞ; nðKÞ;SnðLÞ; nðLÞ and nðKÞ; nðLÞ-N as n-N (see (2.87)),the asymptotic formula (i) for K and L instead of J; (2.87), (2.14), ZðLÞ ¼ 1 ZðKÞ(see (2.84)), (2.15), Wo1oi; (2.87) and (2.86), and (2.3) we arrive at the followingcontradiction, where k40; and thus conclude the proof of (2.84):

Fn ¼Z

K

minpASnðKÞ


þZ

L

minpASnðLÞ


\ div

Z aþdd

K

f1

nðKÞ1d

0@

1Aþ

Z aþdd

L

f1

nðLÞ1d

0@

1A

0@

1A

B div

Z aþdd

J

ZðKÞaþd

d f1

ðWZðKÞnÞ1d

0@

1Aþ ZðLÞ

aþdd f

1

ðiZðLÞnÞ1d

0@

1A

0@

1A

B div

Z aþdd

J

ZðKÞ f1

ðWnÞ1d

0@

1Aþ ð1 ZðKÞÞ f

1

ðinÞ1d

0@

1A

0@

1A

X ð1 þ kÞ div

Z aþdd

J

f1

ðZðKÞWn þ ð1 ZðKÞÞinÞ1d

0@

1A

B ð1 þ kÞ div

Z aþdd

J

f1

ðnðKÞ þ nðLÞÞ1d

0@

1ABð1 þ kÞ div

Z aþdd

J

f1

n1d

!

B ð1 þ kÞ Fn as n-N in the subsequence from ð2:87Þ:

Applying (2.83) and (2.84) to measurable KCJ yield the following:

#ðK-SnÞpnðKÞ ¼ oðnÞ as n-N for oMðKÞ ¼ 0;

#ðK-SnÞ ¼ nðKÞ oðnÞBZðKÞn as n-N for 0ooMðKÞpoMðJÞ:


This concludes the proof of (2.82). A similar, even simpler proof shows that

%Sn is uniformly distributed in J with respect to the density wd=ðaþdÞ

as n-N: ð2:88Þ

2.8. Conclusion

Having proved (2.58) and (2.70), (2.19) and (2.50), and (2.82) and (2.88), the proofof Theorem 1 is complete.

3. Distortion of high resolution vector quantization

3.1. Distortion of vector quantization

Let C1;y;Cn be n measurable sets which tile a measurable set C in Ed and let

c1;y; cnAEd : To each signal xAC assign the codevector or codeword ci from thecodebook fc1;y; cng; where xACi: (In case of ambiguity, which can occur only for x

in a set of measure 0; choose for ci any codevector where xACi:) Common measuresfor the quality of the thus defined encoder or (vector) quantizer on C can be described

as follows: let f : ½0;þNÞ-½0;þNÞ with f ð0Þ ¼ 0 be increasing, jj � jj a norm on Ed ;

and w : C-Rþ continuous. Up to normalization, w is the density of the source whichgenerates the signals x: Then the corresponding (average) distortion of our quantizeris defined to be

Xn

i¼1

ZCi

f ðjjci ujjÞwðuÞ du:

How should the cells Ci and the codevectors ci be chosen in order to minimize thedistortion? Given c1;y; cn; the distortion is minimized if C1CD1;y;CnCDn whereD1;y;Dn are the Dirichlet–Voronoi cells in C corresponding to c1;y; cn and usingjj � jj: Then the distortion is

ZC

mini¼1;y;n

f f ðjjci ujjÞgwðuÞ du:

Thus the minimum distortion is given by

inf%SCC# %S¼n

ZC

mincA %S

f f ðjjci ujjÞgwðuÞ du

� �:



Clearly, Theorem 2 translates into the following result.

Theorem 3. Let f : ½0;þNÞ-½0;þNÞ satisfy the growth condition (2.1), let

a40 be the corresponding constant, and let jj � jj be a norm on Ed : Then there is a

constant div40; depending only on f and jj � jj; such that the following statement

holds: let CCEd be compact and measurable with jCj40 and w : C-Rþ continuous.

Then

(i) the minimum distortion in the above sense of a quantizer on C with source density w

with n codewords is asymptotically equal to

div

ZC

wðuÞd

aþd du

� �aþdd

f 1

n1d

!as n-N:

If, in addition, C is connected and has positive density, i.e. satisfies (2.2), then the

following claims hold: let ðSnÞ be a sequence of codebooks in C; where #Sn ¼ n for

n ¼ 1; 2;y; and such that the infimum in (i) is attained for Sn: Then


(iii) Sn is uniformly distributed in C with density wd=ðaþdÞ as n-N:

Remark. Theorem 1 admits an analogous interpretation.

4. Error of approximation of probability measures by discrete measures

4.1. Quantization of probability measures

Let f : ½0;NÞ-½0;NÞ be monotone increasing, such that f ðtÞ ¼ 0 only for t ¼ 0

and let jj � jj be a norm on Ed : Then we can define on the space of all probability

measures on Ed the following notion of distance:

r f ðP;QÞ ¼ inf

ZEd�Ed

f ðjjx yjjÞ dmðx; yÞ� �

for probability measures P;Q;

where the infimum is extended over all Borel probability measures m on Ed � Ed

which satisfy the following condition:

PðBÞ ¼ mðB � EdÞ;QðCÞ ¼ mðEd � CÞ for all Borel sets B;CCEd :


r f is a slight extension of the Wasserstein- or Kantorovich metric on the space of all

probability measures on Ed : The question arises as to how well can probabilitymeasures be approximated by discrete measures and to describe the bestapproximating discrete measures. Let Dn denote the space of all discrete probability

measures on Ed the support of which consists of at most n points, n ¼ 1; 2;y: Then

for the error of best approximation of a probability measure P on Ed by probabilitymeasures from Dn the following representation holds:

r f ðP;DnÞ ¼ infDADn

fr f ðP;DÞg ¼ infSCEd

#S¼n

ZEd

minsAS

f f ðjju sjjÞgdPðuÞ� �

:

For this and for references compare Graf and Luschgy [20].

4.2. Asymptotic approximation results

Considering the above, it is clear that Theorem 2 yields the following result onapproximation of probability measures.

Theorem 4. Let f : ½0;þNÞ-½0;þNÞ satisfy the growth condition (2.1), let a40 be

the corresponding constant, and let jj � jj be a norm on Ed : Then there is a constant

div40; depending only on f and jj � jj; such that the following statement holds: let

JCEd be compact and measurable with jJj40 and P a probability measure on J with

continuous density p : J-Rþ: Then

(i) the error of best approximation of P by probability measures from Dn is

asymptotically equal to

div

ZC

pðuÞd

aþd du

� �aþdd

f1

n1d

!as n-N:

If, in addition, J is connected and has positive density, i.e. satisfies (2.2), then the

following claims hold: let ðSnÞ be a sequence of supports in J; where #Sn ¼ n for

n ¼ 1; 2;y; and such that for suitable corresponding discrete probability measures in

Dn the minimum error is attained. Then


(iii) Sn is uniformly distributed in J with density wd=ðaþdÞ as n-N:

Remark. Obviously, Theorem 1 yields a similar result for the approximation ofprobability measures by discrete probability measures on Riemannian manifolds.


5. Error of numerical integration formulae

5.1. Error of numerical integration

Let JCEd be a compact measurable set in Ed with jJj40; let F be a class of

Riemann integrable functions g : J-R and w : J-Rþ a continuous (weight)function. For given sets of n nodes N ¼ fp1;y; pngCJ and n weights W ¼fw1;y;wngCR the error of the numerical integration formula

ZJ

gðuÞwðuÞ duEXn

i¼1

gðpiÞwi for gAF;

is defined by

EðF;w;N;WÞ ¼ supgAF

ZJ

gðuÞwðuÞ du Xn

i¼1

gðxiÞwi

��

( ): ð5:1Þ

The minimum error is then

EðF;w; nÞ ¼ infNCJ;#N¼n

WCR;#W¼n

fEðF;w;N;WÞg: ð5:2Þ

Now the problems arise to determine EðF;w; nÞ and to describe the optimal choicesof nodes and weights. While for general F not much can be said, we consider a spaceof functions for which rather precise information can be given.


A modulus of continuity is an increasing continuous function f : ½0;þNÞ-½0;þNÞwith f ð0Þ ¼ 0 such that

f ðs þ tÞpf ðsÞ þ f ðtÞ for s; tX0: ð5:3Þ

Given a modulus of continuity f and a norm jj � jj on Ed ; define the corresponding

Holder class of functions for JCEd by

H f ðjj � jj; JÞ ¼ fg : J-R : jgðuÞ gðvÞjpf ðjju vjjÞ for u; vAJg: ð5:4Þ

Theorem 5. Let f : ½0;þNÞ-½0;þNÞ be a modulus of continuity which satisfies the

growth condition (2.1), let a40 be the corresponding constant, and let jj � jj be a norm

on Ed : Then there is a constant div40; depending only on f and jj � jj; such that we have


the following: let JCEd be compact and measurable with jJj40 and w : J-Rþ

continuous. Then

(i) for the minimum error of a numerical integration formula with weight function w

for the Holder class H f ¼ H f ðJ; jj � jjÞ holds the asymptotic formula,

EðH f ;w; nÞBdiv

ZJ

wðuÞd

aþd du

� �aþdd

f 1

n1d

!as n-N:

If, in addition, J is connected and has positive density, i.e. satisfies (2.2), then the

following assertions hold: let ðNnÞ be a sequence of sets of nodes in J; where #Nn ¼ n

for n ¼ 1; 2;y; and such that the infimum in (4.2) is attained for Nn and suitable

corresponding Wn: Then

(ii) there is a constant d41 such that Nn is ð1=dn1=d ; d=n1=dÞ-Delone in J for

n ¼ 1; 2;y;

(iii) Nn is uniformly distributed in J with density wd=ðaþdÞ as n-N:

Remark. An analogous result holds for Riemannian d-manifolds instead of Ed andjj � jj: The asymptotic formula (5.3) for d ¼ 2; jj � jj ¼ jj � jj2 and w ¼ 1 is due to

Babenko [2]. For general d and arbitrary norm a related yet different asymptoticformula was given by Chernaya [8]. The fact that in Chernaya’s result w is assumedto be Lebesgue integrable can be reduced to the continuous case by Lusin’s theorem.A pertinent idea is described also by Sobol’ [45].

5.3. Proof of Theorem 5

First, the following will be shown:

Let N ¼ fp1;y; pngCJ and let h : J-R be defined by

hðuÞ ¼ minpAN

f f ðjjp ujjÞg: Then hAH f : ð5:5Þ

To see this, let u; vAJ: By exchanging u and v; if necessary, we may assume thathðuÞXhðvÞ: Then

0p hðuÞ hðvÞ ¼ minpAN

f f ðjjp ujjÞg minqAN

f f ðjjq vjjÞg

¼ f ðjjp ujjÞ f ðjjq vjjÞ for suitable p; qAN


p f ðjjq ujjÞ f ðjjq vjjÞ

¼ f ðjjq v þ v ujjÞ f ðjjq vjjÞ

p f ðjjq vjj þ jjv ujjÞ f ðjjq vjjÞ

p f ðjjq vjjÞ þ f ðjjv ujjÞ f ðjjq vjjÞ ¼ f ðjju vjjÞ;

where we have used the assumptions that f is increasing and that is a modulus ofcontinuity.

Secondly, let N ¼ fp1;y; pngCJ and W ¼ fw1;y;wngCR: Define Di ¼ fuAJ :

jjpi ujjr jjpj ujj for j ¼ 1;y; ng; Ei ¼ Di\ðD1,y,Di1Þ; %wi ¼R

EiwðuÞ du;

i ¼ 1;y; n: %W ¼ f %w;y; %wng:Then

EðH f ;w;N;WÞXEðH f ;w;N; %WÞ ¼Z

J

minpAN

f f ðjjp ujjÞgwðuÞ du: ð5:6Þ

Since J is the disjoint union of E1;y;En; gAH f ;EiCDi; the assumptions that f is

increasing and wX0; the definition of h; hAH f and hðpiÞ ¼ 0 (see (5.5)) togetherimply (5.6):

EðH f ;w;N; %WÞ ¼ supgAH f

ZJ

gðuÞwðuÞ du Xn

i¼1

gðpiÞZ

Ei

wðuÞ du

��

( )

p supgAH f

Xn

i¼1

ZEi

jgðuÞ gðpiÞjwðuÞ du

( )

pXn

i¼1

ZEi

f ðjjpi ujjÞwðuÞ du ¼Xn

i¼1

ZEi

minpjAN

f f ðjjpj ujjÞgwðuÞ du

¼Z

J

minpAN

f f ðjjp ujjÞgwðuÞ du ¼Z

J

hðuÞwðuÞ du

¼Z

J

hðuÞwðuÞ du Xn

i¼1

hðpiÞwipEðH f ;w;N;WÞ:

Thirdly, it follows from (5.6) that

EðH f ;w; nÞ ¼ infNCJ;#N¼n

WCR;#W¼n

fEðH f ;w;N;WÞg ¼ infNCJ#N¼n

fEðH f ;w;N; %WÞg

¼ infNCJ#N¼n

ZJ

minpAN

f f ðjjp ujjÞgwðuÞ du

� �:

Now apply Theorem 2 to obtain Theorem 5.


6. Volume approximation of convex bodies

6.1. Best volume approximation of convex bodies

Let C be a convex body in Ed ; i.e. a compact convex subset of Ed with non-emptyinterior and let Pc

ðnÞ be the set of all convex polytopes with n facets which are

circumscribed to C: Let dV ð�; �Þ denote the symmetric difference metric on the space

of all convex bodies in Ed : The problems arise to determine or estimate thequantity

dV ðC;PcðnÞÞ ¼ inffdV ðC;PÞ : PAPc

ðnÞg

and to describe the polytopes PAPcðnÞ for which the infimum is attained, the best

approximating polytopes of C in PcðnÞ with respect to dV : See [25,27] for the literature

on this and related problems.

6.2. Asymptotic approximation results

Refining a result of Boroczky [4] (asymptotic formula), the following result hasbeen proved by Gruber [31] (asymptotic formula and Delone property) andGlasauer and Gruber [19] (uniform distribution property). A weaker version of it(without the error term in the asymptotic formula) can be obtained as a simpleconsequence of Theorem 1 above and its proof; compare the correspondingargument in [30].

Theorem 6. There is a constant divd140 with the following property: let C be a

convex body in Ed (with boundary) of class C3 and with Gauss curvature kC40: Let

ðPnÞ;PnAPcðnÞ; be a sequence of best approximating polytopes of C: Then we have the

following statements:

(i) dV ðC;PcðnÞÞB1

2divd1

Zbd C

kCðxÞ1

dþ1 dsðxÞ� �dþ1

d11

n2

d1

þ O 1

n2

d1þ 13ðd1Þe

0@

1A

as n-N; for any e40;

(ii) there is a constant d41 such that the inradius of each facet of Pn is at least

1=dn1=ðd1Þ and the circumradius at most d=n1=ðd1Þ for n ¼ d þ 1; d þ 2;y;(iii) the sets C-bd Pn are uniformly distributed on bd C; endowed with the ordinary

surface area measure s; with respect to the density k1=ðdþ1ÞC as n-N:

Remark. For d ¼ 3; a sharper result was given in [32]. See [24,26] for earlierpertinent results.


7. The isoperimetric problem for convex polytopes in Minkowski spaces

7.1. The isoperimetric problem in Minkowski spaces

Let Ed be endowed with an additional norm. While in this new normed space thenatural notion of volume is the ordinary volume Vð�Þ or j � j (Haar measure), severaldifferent proposals for surface area have been made by Busemann [6], Holmes andThompson [35] and Benson [3]. These amount to the introduction of an o-symmetricconvex body I ; the isoperimetrix, depending in a suitable way on the solid unit ball ofthis norm. The surface area SIðCÞ of a convex body C then is defined to be

SI ðCÞ ¼ limd-þ0

VðC þ dIÞ VðCÞd

; where C þ dI ¼ fx þ dy : xAC; yAIg:

See the book [47]. For n ¼ d þ 1; d þ 2;y; let Pn be a convex polytope which hasminimum isoperimetric quotient

SI ðPnÞd

VðPnÞd1

among all convex polytopes in Ed with n facets. The problem arises to determine orestimate the minimum isoperimetric quotient and to describe the form of theminimizing polytopes Pn:

7.2. Asymptotic isoperimetric results

A result of Diskant [10,11], which generalizes a theorem of Lindelof [38] (alsoproved by Minkowski [40]) for the Euclidean case, says that after a suitablehomothety has been applied to Pn; we may assume that Pn is circumscribed to I ; i.e.PnAPc

ðnÞðIÞ: Then it is easy to see that

SIðPnÞd

VðPnÞd1¼ ddVðPnÞ:

Since the isoperimetric quotient is minimal for Pn; the polytope Pn

thus is best approximating of I in PcðnÞðIÞ: Hence Theorem 5 yields the following

result.

Theorem 7. There is a constant divd140 such that the following hold: let I be an

isoperimetrix in Ed of class C3 with kI40: For n ¼ d þ 1; d þ 2;y; let Pn be a convex


polytope with n facets circumscribed to I and minimum isoperimetric quotient

SI ðPnÞd=VðPnÞd1: Then the following assertions hold:

(i) SI ðPnÞd

VðPnÞd1B ddVðIÞ þ dd

2divd1

Zbd I

kCðxÞ1

dþ1 dsðxÞ� �dþ1

d1 1

n2

d1

þ O1

n2

d1þ 1

3ðd1Þe

!as n-N; for any e40;

(ii) there is a constant d41 such that the inradius of each facet of Pn is at least

1=dn1=ðd1Þ and the circumradius at most d=n1=ðd1Þ for n ¼ d þ 1; d þ 2;y;(iii) the sets C-bd Pn are uniformly distributed on bd C; endowed with the ordinary

surface area measure s; with respect to the density k1=ðdþ1ÞC as n-N:

Remark. For d ¼ 3; a sharper result was given in [32].

Acknowledgments

For their helpful remarks in the preparation of this article and references to theliterature I am indebted to Professors Paul Goodey, Siegfried Graf, Robert Gray andFranz Schnitzer.

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