optimal configurations of finite sets in riemannian 2-manifolds

Optimal Con¢gurations of Finite Sets inRiemannian 2-Manifolds

PETER M. GRUBERAbteilung. f. Analysis, Techn. Univ. Wien, Wiedner Hauptstr. 8-10/1142, A-1040 Vienna,Austria. e-mail: [email protected]

(Received: 16 November 1999)

Communicated by R. Schneider

Abstract. In this article we ¢rst extend Fejes Toth's basic result on sums of moments from theplane toRiemannian2-manifolds.The extension aswell as theoriginal result show that for certaingeometric or analytic problems regular hexagonal arrangements are optimal or almost optimal.Then a stability counterpart of the moment theorem for Riemannian 2-manifolds is given. Itshows that, conversely, for several problems the optimal con¢gurations are approximately regularhexagonal. Finally, the following applications are considered: (i) Description of the form of bestapproximating convex polytopes circumscribed to smooth convex bodies in E3 as the numberof facets tends to in¢nity. (ii) Description of the convex polytopes with minimum isoperimetricquotient in three-dimensional Minkowski spaces as the number of facets tends to in¢nity. (iii)Description of the arrangement of nodes in optimal numerical integration formulae for Ho« lderclasses of functions of two variables as the number of nodes tends to in¢nity.

Mathematics Subject Classi¢cations (2000). Primary 52C99, 53C20. Secondary 52A27,52B60, 65D30.

Key words. sums of moments, stability, almost regular hexagonal con¢gurations, best approxi-mating polytopes, minimum isoperimetric quotient, optimal integration formulae.

1. Introduction

1.1. Amongst a small group of general results in discrete geometry is the followingtheorem of Laszlo Fejes Toth on sums of moments : let f : �0;�1� ! �0;�1� benondecreasing and let H be a convex 3; 4; 5; or 6-gon in the Euclidean plane E2.Then, for any ¢nite set S in E2,Z

Hminp2Sf f �kpÿ xk�g dxX n

ZHn

f �kxk� dx; �1:1�

where n � #S is the number of points of S andHn is a regular hexagon in E2 of areajHj=n and center at the origin o ; k � k and j � j denote the Euclidean norm and theordinary area measure in E2. The inequality (1.1), resp. a slightly different versionof it, was ¢rst proved by Laszlo Fejes Toth [8] for the 2-sphere instead of E2,and only later for E2, see [9]. For alternative proofs, in some cases for surfaces

Geometriae Dedicata 84: 271^320, 2001. 271# 2001 Kluwer Academic Publishers. Printed in the Netherlands.

of constant curvature and also for special nonmonotone functions, see LaszloFejes Toth [10], Imre [24], Bollobas and Stern [2], Gabor Fejes Toth [7],Florian [12], Papadimitriou [27], Haimovich and Magnanti [21] and, more recently,Bo« ro« czky and Ludwig [3] and the author [19].

Recent applications of the estimate (1.1) to asymptotic approximations of convexbodies inE3 by polytopes are due to Gruber [14] and Bo« ro« czky and Ludwig [3]. Otherapplications deal, for example, with planar packing and covering problems, prob-lems of quantization of data, Gauss channels, optimal location in economics,the isoperimetric problem for special classes of convex polytopes in E3, the optimalchoice of nodes in numerical integration formulae for functions of two variablesand game theory. For references see [20]. Presumably, Fejes Toth's theorem canbe applied also in biology.

Surprisingly, the inequality (1.1) may be arbitrarily weak: for suitable f thequotient of the left and right sides in (1.1) has no ¢nite upper bound, even if Sis chosen so as to minimize the left-hand integral.

1.2. In view of the applications, two extensions of (1.1) suggest themselves, namely tohigher dimensions and to Riemannian 2-manifolds. While the former at presentseems to be out of reach, it is possible to extend (1.1) to Riemannian 2-manifolds.To enlarge the range of applications, we consider functions f and Riemannian2-manifoldsM with metric RM which satisfy rather mild conditions where, of course,functions as in the last paragraph have to be excluded, see 2.1.

Let J be a Jordan measurable set inM and for n � 1; 2; . . . let Hn denote a regularhexagon of E2 of area equal to the Riemannian area of J over n and with center o.Then

infS�M

#SW n

ZJ

minp2Sf f �RM� p; x��g doM�x� � n

ZHn

f �kxk� dx as n!1:

As is the case for (1.1), also this result implies that for a series of geometric andanalytic problems the regular hexagonal arrangements are close to optimal or evenoptimal. The more dif¢cult question asks whether, in such situations, optimaland almost optimal con¢gurations are almost regular hexagonal. As expected,the answer is af¢rmative and is a consequence of the following weak stabilitycounterpart of the above result: LetZ

Jminp2Sn

ff �RM�p; x��g doM�x� � nZHn

f �kxk� dx as n!1;

where �Sn� is a sequence of ¢nite sets in M with #Sn W n. Then Sn is uniformly dis-tributed in J and asymptotically a regular hexagonal pattern (for de¢nitions, see3.1). In case f �t� � ta, where 0 < aW 2, these results were announced by the author[19] and Gabor Fejes Toth has informed us of the proof of a similar stability resultfor the Euclidean plane.

272 PETER M. GRUBER

1.3. We consider several applications of the above results:(i) Approximation of convex bodies. A convex body in E3 is a compact convex set

in E3 with nonempty interior. Let dV �: ; :� denote the symmetric difference metric onthe space of all convex bodies in E3. Given a convex body C, let Pc

�n� be the set of allconvex polytopes circumscribed to C and having at most n facets. Anintensively studied topic is the investigation of the quantity dV �C;Pc

�n�� inffdV �C;P� : P 2 Pc

�n�g and the description of the polytopes Pn 2 Pc�n� for which

the in¢mum is attained, that is, the best approximating polytopes of C in Pc�n�;

see [16, 17]. Much is known about the asymptotics of dV �C;Pc�n�� as n!1, but

a precise determination of dV �C;Pc�n�� and description of Pn seems impossible at

present. In this article it will be shown that for a suf¢ciently smooth convex bodyC in E3 the best approximating polytopes of C in Pc

�n� have asymptotically regularhexagonal facets (with respect to a suitable Riemannian metric) as n tends to in¢nity.

(ii) The isoperimetric problem in Minkowski space. Let Ed be endowed with afurther norm. In this new normed space the volume V �:� is the ordinary volumein Ed , but there are different notions of surface area, e.g. those proposed byBusemann and by Holmes and Thompson. These notions amount to the introductionof an o-symmetric convex body I , the isoperimetrix, such that the surface area of aconvex body C is de¢ned by SI �C� � lim�V �C � dI� ÿ V �C��=d as d!�0. Inthe Euclidean case I is the Euclidean unit ball; see [31]. With the help of a resultof Diskant [5, 6] we will prove that the convex polytopes in E3 with minimumisoperimetric quotient amongst those with n facets have asymptotically regular hex-agonal facets (with respect to a suitable Riemannian metric) as n tends to in¢nity.

(iii) Numerical integration. Let M and J be as above and consider a class F ofRiemann integrable functions g : J ! R. For given sets of nodesNn � f p1; . . . ; png in J and weightsWn � fw1; . . . ;wng inR let the error and themini-mum error be de¢ned as follows

E�F ;Nn;Wn� � supg2F

ZJg�x�doM�x� ÿ

Xnk�1

wk g�pk��

��( )

;

E�F ; n� � infNn;WnfE�F ;Nn;Wn�g:

The problem arises to determine E�F ; n� and to describe the optimal choices of nodesand weights. For general F (and for M of dimension greater than 2) one cannotexpect much, but in the following case some information can be given: considerfor a modulus of continuity f : �0;1� ! �0;1� (see 6.1) which satis¢es a weakgrowth condition (see 2.1), the Ho« lder class Hf of all functions g : J ! R such thatjg�x� ÿ g�y�jW f �RM�x; y�� for all x; y 2 J. Using the above results and differentialgeometric tools we will show that

E�Hf ; n� � nZHn

f �kxk�dx as n!1

FINITE SETS IN RIEMANNIAN 2-MANIFOLDS 273

and, if �Nn� and �Wn� are sequences of nodes and weights, respectively, such thatE�Hf ;Nn;Wn� � E�Hf ; n� as n!1, then Nn is uniformly distributed in J andasymptotically a regular hexagonal pattern.

The asymptotic formula for M � E2 and S2 is, in essence, due to Babenko [1].

2. Extension of Fejes Toth's Theorem to Riemannian 2-Manifolds

2.1. THE RESULT

A function f : �0;�1� ! �0;�1� satis¢es the growth condition if f �0� � 0; f is con-tinuous, strictly increasing and for each a > 1 there are b > g > 1 (and vice versa)such that gf �t�W f �at�W bf �t� for all suf¢ciently small tX 0. Here b and g maybe chosen arbitrarily close to 1 if a is suf¢ciently close to 1 (and vice versa).

Let M be a Riemannian 2-manifold. By this we mean that M is a 2-manifold M ofdifferentiability class C1 with a metric tensor ¢eld having continuous coef¢cients. LetRM and oM denote the corresponding Riemannian metric and area measure. A set Jin M is Jordan measurable if it has compact closure cl J and its boundary bd Jhas measure 0; see also 2.2.1.

THEOREM 1. Let f : �0;�1� ! �0;�1� satisfy the growth condition and let J be aJordan measurable set in M with oM�J� > 0.Then

infS�M

#SW n

ZJ

minp2Sff �RM�p; x��gdoM�x� � n

ZHn

f �kxk�dx as n!1; �2:1�

where Hn is a regular hexagon in E2 with area oM�J�=n and center o. The formula(2.1) continues to hold under the assumption that S is contained in J.

It follows from the proof that, in principle, it is possible to construct a sequence ofsets S � Sn which satisfy the asymptotic relation (2.1). In a more general versionof the formula (2.1) there is a continuous weight factor under the left integral.In its proof the weight factor is used to de¢ne a new metric tensor ¢eld on M. ThenTheorem 1 can be applied.

The proof of Theorem 1 is split into two parts.

2.2. PREPARATIONS FOR THE PROOF

2.2.1. For a better understanding we ¢rst describe the de¢nitions of RM;oM and ofJordan measurability. Let U be an open neighborhood of a point in M with cor-responding homeomorphism h which maps U onto an open set U 0 in E2. For eachu 2 U let qu� : � be the corresponding positive de¢nite quadratic form on E2. A curveK in U is of class C1 if it has a parametrization x : �a; b� ! U such that u � h � x is a

274 PETER M. GRUBER

parametrization of a curve of class C1 in U 0. The length of K then is de¢ned asZ b

aq1=2u�s��_u�s��ds:

By means of (an appropriate) dissection and addition, the notion of length can bede¢ned for curves in M which are not contained in a single neighborhood. Forx; y 2M let their Riemannian distance RM�x; y� be the in¢mum of the lengths ofcurves of class C1 in M connecting x and y. A set J in U is Jordan measurableif h�J� is Jordan measurable as a subset ofE2. Then its area measure oM�J� is de¢nedas Z

h�J��det qu�1=2 du:

Again, by dissection and addition, we can de¢ne the concepts of Jordanmeasurability and area measure for sets in M which are not contained in a singleneighborhood. This de¢nition of Jordan measurability and the one in 1.2 areequivalent.

2.2.2. The moments M�D; p� and M�a; v�. If D is a compact convex set in E2, itsmoment M�D; p� with respect to a point p in E2 is de¢ned asZ

Df �kpÿ xk� dx: �2:2�

Write M�a; v� for the moment of a regular v-gon of area a with respect to its center.A suitable linear transformation together with (1.1) yields the following.

Let q� : � be a positive de¢nite quadratic form on E2 and Q � E2 a convex 3; 4; 5,or 6-gon. Then for m � 1; 2; . . . ; and any set S in E2 with #SWm holdsZ

Qminr2Sff �q1=2�rÿ s��g ds �det q�1=2 XmM

jQj�det q�1=2m

; 6�:

�2:3�

The growth condition (1.2) together with an argument of Babenko [1] and the appli-cation of a suitable linear transformation implies the next result.

Let l > 1; q� : � a positive de¢nite quadratic form on E2, and R a Jordanmeasurable set in E2. Then for each su¤ciently large m there is a set T � Rwith #T Wm with the following properties:

(i) The distance (with respect to the norm q� : �1=2) of any point of R to thenearest point of T is O�mÿ1=2�.(ii)

ZR

minr2Tff �q1=2�rÿ s��g ds �det q�1=2 W lmM

jRj�det q�1=2m

; 6�: �2:4�


2.2.3. Properties of M�a; 6�. A special case of a result of the author [19] says that

M�a; 6� is convex for aX 0: (2.5)

Clearly,

M�0; 6� � 0 and M�a; 6� is strictly increasing for aX 0: (2.6)

As a consequence of (2.5) we will show that

mMam; 6

� �is nonincreasing for all a > 0 and mX 1: (2.7)

Let g be a convex nondecreasing function for tX 0 with g�0� � 0. An inspection ofthe graph of g shows that g�t�=t is nondecreasing for t > 0. Hence tg�1=t� isnonincreasing for t > 0. Applying this with g�t� �M�at; 6�, where we take intoaccount (2.5), (2.6), yields (2.7).

Since f satis¢es the growth condition, the de¢nition of M�a; 6� implies that

M�a; 6� satis¢es the growth condition. (2.8)

2.2.4. The Dirichlet^Voronoi cells DM�S; p�. For a ¢nite set S �M let

DM�S; p� � fx 2 J : RM�p; x�W RM�q; x� for each q 2 Sg; p 2 S:

Then the following holds:

Let �Sn� be a sequence of ¢nite sets in M such that #SW n andZJ

minp2Sn

ff �RM�p; x��gdoM�x� ! 0 as n!1:

Then maxp2Sn

fdiam�DM�Sn; p� \ intJ�g ! 0 as n!1: �2:9�

Here diam stands for diameter. For the proof of (2.9) it is suf¢cient to show thefollowing statement: let E > 0, then for all suf¢ciently large n: given p 2 Sn andx 2 DM�Sn; p� \ int J, then RM�p; x� < 2E. The compactness of cl int J implies thatthere are ¢nitely many points in int J such that the E-neighborhoods of these pointscover int J. If n is suf¢ciently large, each of these E-neighborhoods contains at leastone point of Sn; otherwise the integral in (2.9) cannot converge to 0. Consequently,for each x 2 int J the nearest point of Sn has distance less than 2E from x. For eachx 2 DM�Sn; p� \ int J the point p is (one of) the nearest point(s) of Sn, we thus havethat RM�p; x� < 2E. This concludes the proof of the above statement and thus of (2.9).

The next remark is obvious.

276 PETER M. GRUBER

Let S �M be ¢nite and K � J Jordan measurable. De¢neS�K� � fp 2 S : int �DM�S; p� \ K� 6� ;g; n�K� � #S�K�. ThenZ

Kminp2Sff �RM�p; x��gdoM�x� �

ZK

minp2S�K�

ff �RM�p; x��gdoM�x�: �2:10�

2.2.5. In this subsection we use a system of (small) neighborhoods in M to transferinformation from E2 to J and vice versa. Let l > 1. For each p 2M chooseU; h �'' 0 '' and U 0 � h�U� as in 2.2.1, where the neighborhood U is so small thatfor the positive quadratic form q� : � corresponding to p and h the followinginequalities hold; here qu� : � is the positive quadratic form corresponding tou 2 U and h.

1lq1=2�x0 ÿ y0�W RM�x; y�W lq1=2�x0 ÿ y0� for x; y 2 U;

1l�det q�1=2 W �det qu�1=2 W l�det q�1=2 for u 2 U;

1ljK 0j�det q�1=2 WoM�K�W ljK 0j�det q�1=2 for Jordan measurable K � U :

Let V be a Jordan measurable open neighborhood of p with clV � U . As p rangesover the compact set cl J the neighborhoods V form an open covering of clJ. Thusthere is a ¢nite subcover. Therefore we may choose points pl 2 J; l � 1; . . . ; k, say,and corresponding neighborhoods Ul;Vl , homeomorphisms hl �'' 0 '', and positivequadratic forms ql such that

1lq1=2l �x0 ÿ y0�W RM�x; y�W lq1=2l �x0 ÿ y0� for x; y 2 Ul; (2.11)

1l�det ql�1=2 W �det qu�1=2 W l�det ql�1=2 for u 2 Ul; (2.12)

1ljK 0j�det ql�1=2 WoM�K�W ljK 0j�det ql�1=2 for Jordan measurableK � Ul :

(2.13)

Consider the sets Il � J \ �Vl n �V1 [ . . . [ Vlÿ1�� for l � 1; . . . ; k: Since each Vl isJordan measurable, clVl � Ul and J � V1 [ . . . [ Vl , we see that

J is the disjoint union of the Jordan measurable setsIl � Vl � Ul; l � 1; . . . ; k: (2.14)

2.3. PROOF OF THEOREM 1

2.3.1. For the proof of Theorem 1 two estimates are needed. The ¢rst one is thefollowing, the second one will be given in 2.3.2.


Let 0 < a < 1. Then for all su¤ciently large n holds

infS�M

#SW n

ZJ

minp2Sff �RM�p; x�gdoM�x�X a nM

oM�J�n

; 6� �

: �2:15�

By the growth condition for f and (2.8) we may choose l > 1 and a correspondingm > 1 so that

Mal; 6

� �X

1mM�a; 6� for all su¤ciently small aX 0,

ftl

� �X

1mf �t� for all su¤ciently small tX 0 and such that

1l2m3

> a: (2.16)

For this l choose pl; hl � '' 0 '';Ul;Vl; Il; ql; l � 1; . . . ; k, as described in 2.2.5. Then,in particular, (2.11)-(2.14) hold. Next, choose for each l sets Qli; i � 1; . . . ; il , withthe following properties:

(i) The sets Qli; i � 1; . . . ; il, are compact and pairwise disjoint sets inintIl � Ul .(ii) Q0li is a square in E2 of edgelength 1=kl , say, (with respect to the norm

q1=2l � : �), where kl is a positive integer. Clearly, Qli is Jordan measurable.

(iii)Xl;i

oM�Qli�X 1loM�J�. (2.17)

For n � 1; 2 . . ., choose sets Sn in M with #Sn W n such that

infS�M

#SW n

ZJ

minp2Sff �RM�p; x�g doM�x�X 1

l

ZJ

minp2Sn

ff �RM�p; x��g doM�x�: �2:18�

Such a choice is possible since for any n the left side of (2.18) is positive. De¢neSnli � Sn�Qli� and nli � n�Qli� as in (2.10). Taking into account (2.14) and (2.17i),an application of (2.9) and (2.10) then shows that

for all su¤ciently large n hold:(i) The sets Snli �� Ul�; l � 1; . . . ; k; i � 1; . . . ; il, are pairwise disjoint.(ii)

Xl;i

nli W n.

(iii) minp2Sn

ff �RM�p; x��g � minp2Snli

f f �RM�p; x��g for x 2 Qli. (2.19)

Now, using (2.18), (2.17i), (2.19iii), (2.11), (2.12), the de¢nition of the integral in M,(2.16), (2.3), (2.13), (2.6), (2.5), Jensen's inequality, (2.7), (2.19ii), (2.17iii), (2.6), and

278 PETER M. GRUBER

(2.16), we obtain the estimate (2.15):

infS�M

#SW n

ZJ

minp2Sff �RM�p; x�gdoM�x�

X1l

ZJ

minp2Sn

ff �RM�p; x��gdoM�x�

X1l

Xl;i

ZQli

minp2Snli

ff �RM�p; x��gdoM�x�

X1l

Xl;i

ZQ0li

minr2S0nli

f1lq1=2l �rÿ s�

� �� ds

1l�det ql�1=2

X1l2m

Xl;i

nliMjQ0lij�det ql�1=2

nli; 6

!X

1l2m

Xl;i

nliMoM�Qli�lnli

; 6� �

X1l2m�n11 � � � � � nkik�M

oM�Q11� � � � � � oM�Qkik �l�n11 � � � � � nkik�

; 6� �

X1l2m

nMoM�J�l2n

; 6� �

X1

l2m3nM

oM�J�n

; 6� �

> a nMoM�J�

n; 6

� �for all suf¢ciently large n.

2.3.2 The second estimate we require is as follows:

Let b > 1. Then for all su¤ciently large n holds

infS�J

#SW n

ZJ

minp2Sff �RM�p; x�gdoM�x�W b nM

oM�J�n

; 6� �

: �2:20�

By the growth condition for f and (2.8) we may choose l > 1 and a correspondingm > 1 so that

f �lt�W mf �t� for all sufficiently small tX 0;


M�la; 6�W mM�a; 6�; for all sufficiently small aX 0; and such thatl2m5 < b: (2.21)

For this l choose pl; . . . ; as described in 2.2.5. Next, consider for each l anedge-to-edge tiling of E2 with translates of a closed square of areaPk

l�1 jI 0l j�det ql�1=2�det ql�1=2

n; (2.22)

where n > 0 is so small that each of these k tilings has the following properties: letR0lj; j � 1; . . . ; jl; be the tiles of the lth tiling which meet I 0l , then

I 0l � R0l1 [ . . . [ R0ljl � U 0l ; theR0lj; j � 1; . . . ; jl; have disjoint interiors; (2.23)

jR0l1j � . . . � jR0ljl j; jI 0l jW jl jR0l1jW ljI 0l j: (2.24)

Each Rlj � hÿ1l �R0lj� �� Ul� is Jordan measurable. For the proof that

j0jlW l

Pkl�1 jI 0l j�det ql�1=2jI 0l j�det ql�1=2

; where j0 � j1 � . . .� jk; (2.25)

consider the second inequality in (2.24), multiply both sides by �det ql�1=2, insert thevalue of jR0l1j and sum over all l. This gives j0nW l or nW l=j0. Now considerthe ¢rst inequality in (2.24), insert the value of jR0l1j and note that nW l=j0. Thisimplies (2.25).

Choose m0 so large that for each of the positive de¢nite quadratic forms ql� : � andeach of the Jordan measurable sets �Rlj \ J�0 the conclusion of (2.4) holds for allmXm0. Then, using (2.6) we see that

for each mXm0 there are sets T 0mlj � �Rlj \ J�0 with #T 0mlj Wm where

(i) The distance (with respect to the norm q� : �1=2) of any point of�Rlj \ J�0 from the nearest point of Tmlj is O�mÿ1=2�.

(ii)Z�Rlj\J�0

minr2T 0mlj

ff �q1=2l �rÿ s��gds�det ql�1=2 W lmMjR0ljj�det ql�1=2

m; 6

!: (2.26)

After these preparations for the proof of (2.19), we assume ¢rst that n has the formn � j0m with mXm0. Let Tn �

Sl;j Tmlj, where Tmlj � hÿ1l �T 0mlj� �� Rlj \ J�. Clearly,

Tn � J and #Tn W j0m � n. This together with (2.14), (2.23), Tn �S

l;j Tmlj, (2.12)and the de¢nition of the integral in M, (2.11), the growth condition for f , (2.26),

280 PETER M. GRUBER

(2.21), (2.26ii), (2.24), (2.6), (2.25), (2.6), (2.13), (2.14), and (2.21) implies that thereis an m1 Xm0 such that

infT�J

#T W n

ZJ

minp2Tf f �RM� p; x�g doM�x�

W

ZJ

minp2Tnf f �RM� p; x�g doM�x�

WXl;j

ZRlj\Il

minp2Tmlj

f f �RM� p; x�g doM�x�

W lXl;j

Z�Rlj\Il �0

minr2T 0mlj

f f �lq1=2l �rÿ s��gds �det ql�1=2

W l2mXl;j

mMjR0ljj�det ql�1=2

m; 6

!W l2m

Xl

jl mMljI 0l j�det ql�1=2

jlm; 6

!

W l2mXl

jl mMl2P

l jI 0l j�det ql�1=2j0m

; 6

!W l2m j0 mM

l3oM�J�j0m

; 6

!

W l2m4nMoM�J�

n; 6

� �for all n of the form n � j0m;mXm1. (2.27)

For general n we argue as follows: let m2 Xm1 be so large that �m� 1�=mW l formXm2. Then, given nX j0m2, choose mXm2 such that j0mW n < j0�m� 1�. Com-bining this with (2.27), (2.6), and (2.21) ¢nally yields the estimate (2.20):

infT�J

#T W n

ZJ

minp2Tf f �RM� p; x�gdoM�x�

W infT�J

#T W j0m

ZJ

minp2Tf f �RM� p; x�gdoM�x�

W l2 m4j0 mMoM�J�j0m

; 6� �

W l2m4nMloM�J�

n; 6

� �W l2m5nM

oM�J�n

; 6� �

W b nMoM�J�

n; 6

� �for all nX j0m2:


2.3.4. Theorem 1 is an immediate consequence of (2.15) and (2.20). &

3. Stability of Hexagonal Arrangements in Riemannian 2-Manifolds

3.1. THE STABILITY RESULT

Let M be a Riemannian 2-manifold as de¢ned in 1.2 with Riemannian metric RM ,area measure oM and the concept of Jordan measurability. Let �Sn� be a sequenceof ¢nite sets in M with #Sn W n and #Sn!1 as n!1. Sn is said to be uniformlydistributed in a Jordan measurable set J in M with oM�J� > 0 if, for each Jordanmeasurable set K in J with oM�K� > 0, we have

#�K \ Sn�n

! oM�K�oM�J� as n!1;

see Hlawka [23], p.58. Sn is asymptotically a regular hexagonal pattern of edge lengthRn if the following hold: there exist a positive sequence �Rn� and Landau symbols o�n�and o�1� such that for each point p 2 Sn, with a set of at most o�n� exceptions, therelation

fx : RM�p; x�W 1:1Rng \ Sn � fp; p1; . . . ; p6g; say,

holds, where

RM�p; pj�; RM�pj; pj�1� � �1� o�1��Rn for j � 1; . . . ; 6; p7 � p1:

p1; . . . ; p6 are said to form up to o�1� a regular hexagon with center p of edge lengthRn. (�1� o�1��Rn denotes a quantity between �1ÿ o�1��Rn and �1� o�1��Rn.)

THEOREM 2.Let f : �0;�1� ! �0;�1� satisfy the growth condition, let J be aJordan measurable set in M with oM�J� > 0, and �Sn� a sequence of ¢nite sets inM with #Sn W n for n � 1; 2; . . . ; such thatZ

Jminp2Sn

f f �RM� p; x�gdoM�x� � infS�M

#SW n

ZJ

minp2Sf f �RM� p; x��gdoM�x� as n!1:

�3:1�Then Sn is uniformly distributed in J and is asymptotically a regular hexagonal pat-tern of edge length

2oM�J��3p

n

� �1=2

:

As was the case for Theorem 1, there is also a more general version of Theorem 2with a weight factor in the left integral in (3.1).

After some preparations in 3.2 we ¢rst prove the result for the Euclidean plane in3.3 and then extend it to Riemannian manifolds in 3.4. The former case is more

282 PETER M. GRUBER

dif¢cult, but also the extension has to be done with more care than is usual in ordernot to loose the stability property.

3.2. PREPARATIONS FOR THE PROOF

3.2.1. Simpli¢cations. We note that f , or any continuous strictly increasing functionon �0;1� beween �1ÿ o�t��f �t� and �1� o�t��f �t� as t! 0, satis¢es both the growthcondition and (3.1) (note (2.9)). It therefore suf¢ces to prove Theorem 2 underthe additional assumption that f is continuously differentiable for t > 0. Then

g : �0;�1� ! �0;�1�, de¢ned by g�t� � f �t1=2� for tX 0, is continuouslydi¡erentiable for t > 0 and satis¢es the growth condition. (3.2)

Let G be a primitive of g with G�0� � 0.Positive constants independent of m, respectively n, will be denoted in many cases

just by const. If const appears several times in the same or in consecutive expressions,this does not mean that it denotes necessarily the same constant. To simplify thepresentation, constants are sometimes absorbed by Landau symbols and in someother cases constants are denoted by Greek letters. W always denotes a suitable realwith 0 < W < 1.

A Landau symbol o�1� (and similarly for the other Landau symbols used) is apositive function of m or n which converges to 0 as m!1 or n!1, respectively.If a Landau symbol appears several times in the same or in consecutive expressions,it may be different in each case. Some of our arguments are valid only if m or nis suf¢ciently large. In several cases this is assumed tacitly.

3.2.2. The moment lemma. The following result of Laszlo Fejes Toth [9, 11] will beuseful in the sequel; for the de¢nition of the momentsM�D; p� and M�a; v� see 2.2.2.

Let D be a convex v-gon with area a and let p 2 E2. Then M�D; p�XM�a; v�.(3.3)

3.2.3. Properties of M�a; v�. Set h�a; v� � a=�v tan�p=v�� for aX 0; vX 3. Thenh1=2�a; v� is the inradius of a regular v-gon of area a. Using polar coordinates,we obtain

M�a; v� � 2vZ p

v

0

Z h1=2cosc

0g�r2�rdrdc � v

Z pv

0G

hcos2 c

� �dc: �3:4�

De¢ne M�a; v� for aX 0 and real vX 3 by (3.4).

The ¢rst needed property of M�a; v� was proved in [19]. It says that


M�a; v� is convex for all aX 0; vX 3: (3.5)

We draw two consequences of (3.5):

M�a; v�XM�b; v� �Ma�b; v��aÿ b�;M�a; v� �M�b; v� �Ma�b; v��aÿ b� � 1

2Maa�b� W�aÿ b�; v��aÿ b�2

for all a; bX 0; vX 3: (3.6)

Since g � G0 is continuously differentiable for t > 0 (see (3.2)), the function M�a; v�has continuous partial derivatives up to the second order by (3.4). Now, noting (3.5),we obtain (3.6).

M�a; v� is nonincreasing in v for ¢xed a > 0 and, trivially, strictly increasing in afor ¢xed vX 3. (3.7)

To see this, ¢x a > 0. ThenM�a; v� is convex in v by (3.5). The de¢nition ofM�a; v� in(3.4) together with a simple continuity argument shows that M�a; v� !M�C; o� asv!�1, where C is the (solid) circle of area a and center o. Thus M�a; v� is abounded convex function for vX 3, which implies (3.7).

Second,

const a g�a�WM�a; v�W const a g�a�for all su¤ciently small aX 0 and all vX 3. (3.8)

By (3.7) and its proof, M�C; o�WM�a; v�WM�a; 3�. Since clearly�a=2�g�a=2p�WM�C; o� and M�a; 3�W ag�4a=3 ��

3p �, an application of the growth

condition for g (see (3.2)) yields (3.8).

Third,

const g�a�WMa�a; v�W const g�a�for all su¤ciently small aX 0 and all vX 3. (3.9)

Propositions (3.4) and (3.2) imply that for all suf¢ciently small aX 0 and for vX 3,

Ma � v ha

Z pv

0g

hcos2 c

� �dc

cos2 cW

vv tan p

v

pvg

av tan p

v cos2 pv

� �1

cos2 pvW const g�a�;

Ma X v ha

Z pv

p2v

gh

cos2 c

� �dc

cos2 cX

vv tan p

v

p2v

ga

v tan pv cos2 p

2v

� �1

cos2 p2vX const g�a�:

284 PETER M. GRUBER

Fourth,

Maa�a; v�X constg�a�a

for all su¤ciently small a > 0 and for 5W vW 7: (3.10)

The inequality in (3.10) is a consequence of (3.4) and (3.2):

Maa � v h2a

Z pv

0g0

hcos2 c

� �dc

cos4 cX

vh2a2h

Z pv

0g0

hcos2 c

� �2h sinccos3 c

dc

� 12a tan p

v�g h

cos2 pv

� �ÿ g�h��X const

g�a�a

for all suf¢ciently small aX 0 and for 5W vW 7:

As a preparation for the proof of (3.12) we show that

Maa�a; v�Mvv�a; v� ÿM2av�a; v�

Maa�a; v� X const ag�a�

for all su¤ciently small a > 0; and for 5W vW 7. (3.11)

To see this we ¢rst specify two candidates for an upper bound of Maa, one of whichactually turns out to be an upper bound. We note that (3.2) implies g0X 0. Usingthis and (3.4), we obtain:

Maa � vh2a

Z pv

0g0

hcos2 c

� �dc

cos4 cW

2vh2a

Z pv

p2v

g0h

cos2 c

� �dc

cos4 c; or

2vh2a

Z p2v

0g0

hcos2 c

� �dc

cos4 c

8>>>><>>>>:

9>>>>=>>>>;for all a > 0 and 5W vW 7.

A representation of MaaMvv ÿM2av from [19] and g0X 0 implies that

MaaMvv ÿM2av

� 2p2av5 sin2 p

v

Z pv

0g0

hcos2 c

� �1ÿ sin2 c

sin2 pv

!dc

cos4 c2a cos2 p

v

v sin2 pv

Z pv

0g0

hcos2 c

� �sin2 ccos4 c

dc

X const a2Z p

2v

0g0

hcos2 c

� �dc

cos4 c

Z pv

p2v

g0h

cos2 c

� �dc

cos4 c

for all suf¢ciently small a > 0 and 5W vW 7.


This together with the above upper estimate for Maa and (3.2) then yields (3.11):

MaaMvv ÿM2av

Maa

X

const a2Z p

2v

0g0

hcos2 c

� �dc

cos4 cX const

a2

2h

Z p2v

0g0

hcos2 c


dc; or

const a2Z p

v

p2v

g0h

cos2 c

� �dc

cos4 cX const

a2

2h

Z pv

p2v

g0h

cos2 c


dc

8>>>><>>>>:

9>>>>=>>>>;

X

const gh

cos2 pv

� �ÿ g�h�

� �; or

const a gh

cos2 pv

� �ÿ g

kcos2 p

2v

� �� 8>>>><>>>>:

9>>>>=>>>>;X const ag�a�

for all suf¢ciently small a > 0 and for 5W vW 7.

Fifth,

b2Maa�a; v� � 2bMav�a; v� �Mvv X const a g�a� (3.12)for all su¤ciently small a > 0 and for b 2 R; 5W vW 7:

The polynomial in b in (3.12) attains its minimum for b � ÿMav=Maa. This togetherwith (3.11) implies (3.12):

b2Maa � 2bMav �Mvv XM2

av

Maaÿ 2M2

av

Maa�Mvv �MaaMvv ÿM2

av

MaaX const ag�a�

for all suf¢ciently small a > 0 and for 5W vW 7.

Finally we show, sixth, the following estimate which re¢nes (2.7):

For each s with 0 < s < 1 there is a t > 1 such that for each a > 0 and all suf-¢ciently large m � 1; 2; . . ., holds:

smMasm

; 6� �

X tmMam; 6

� �: �3:13�

Let l � 1=s > 1. Since M�0; 6� � 0 and since M�a; 6� is convex for aX 0by (3.5), an inspection of the graph of M�a; 6� shows that M�a; 6� �

286 PETER M. GRUBER

Ma�a; 6��lÿ 1�aX lM�a; 6�. Hence we have by (3.6), (3.7), (3.8) and (3.10) that

M�la; 6�XM�a; 6� �Ma�a; 6��lÿ 1�a� const M�a; 6��lÿ 1�2

X lM�a; 6� � �lÿ 1�2const M�a; 6�;

or

1lM�la; 6�X 1� �lÿ 1�2

lconst

!M�a; 6� � tM�a; v�; say;

for all suf¢ciently small a > 0.Now replace a by a=m and 1=l by s and multiply both sides with m.

3.2.4. A lower estimate forM�D; o�. The following proposition is a re¢nement of themoment lemma (3.3) for 6-gons. It can be obtained along the lines of Hajos' [22]proof of (3.3) and so we omit its proof.

Let D be a convex 6-gon with jDj � a > 0 and o 2 intD. LetC�d� � C��1ÿ d��a=2 ��

3p �1=2; o�, where 0W d < 1, be the circle of maximum

radius with center o contained in D. Let D�d� denote a convex 6-gon such thatjD�d�j � a; C�d� � D�d�, the vertices of D�d� are equidistant from o, the longestedge of D�d� touches C�d�, and the 5 other edges all have the same length.ThenM�D; o�XM�D�d�; o�. (3.14)

The proof of the next result is surprisingly tedious.

Let D and D�d� be as in (3.14). Then M�D�d�; o�X �1� const d2�M�a; 6� for allsu¤ciently small a > 0 and 0W d < 1. (3.15)

The ¢rst step is to show that

M�D�d�; o� is nondecreasing for 0W d < 1. (3.16)

Since M�D�d�; o� is clearly continuous in d, it is suf¢cient to prove the following:let 0 < d < 1, then for each E < d suf¢ciently close to d we have M�D�E�; o�WM�D�d�; o�. Let p; q; r be consecutive vertices of D�d� such that the line segment�q; r� is the longest edge of D�d�. It is the only edge of D�d� touching C�d� andkpÿ qk < kqÿ rk. Then, if E < d is suf¢ciently close to d, the following hold: ifwe keep all vertices of D�d� ¢xed, except q, and move q parallel to �p; r� a suitable(small) distance in the direction rÿ p, we obtain a convex 6-gon E, say, such thatjEj � a;C�E� � E, precisely one edge of E touches C�E�, and M�E; o� <M�D�d�; o�. (To see the latter notice that E is obtained from D�d� by keeping


D�d� \ E pointwise ¢xed and shearing the triangle D�d� n E parallel to �p; r� onto thetriangle E nD�d�. The shearing strictly decreases distances from o if E is suf¢cientlyclose to d.) By (3.14) M�D�E�; o�WM�E; o�. Since M�E; o� <M�D�d�; o�, this con-cludes the proof of (3.16).

In the second step d is replaced by a parameter j as follows: for 0W d < 1 let0Wj < p=3 be such that the angles under which the longest edge of D�d� andthe 5 other edges appear from o are p=3� 2j and p=3ÿ 2j=5, respectively. Fromnow on we write D�j� rather than D�d�. Let r � r�a;j� be the circumradius ofD�j�. Since the longest edge of D�j� touches the circleC�d� � C��1ÿ d��a=2 ��

3p �1=2; o� and jD�j�j � a, it follows that

�1ÿ d� a

2��3p

� �1=2

� r cosp6� j

� �;

a � r2 sinp6� j

� �cos

p6� j

� �� 5 sin

p6ÿ j

5

� �cos

p6ÿ j

5

� ��

� r212

sinp3� 2j

� �� 52

sinpf 3ÿ 2j

5

� �� :

Hence,

r2 � r2�a;j� � 2a sinp3� 2j

� �� 5 sin

p3ÿ 2j

5

� �� ÿ1;

d � d�j� � 1ÿ �4��3p�1=2 cos

p6� j

� �sin

p3� 2j

� �� 5 sin

p3ÿ 2j

5

� �� ÿ1=2:

A numerical calculation readily implies that 1=4W d0�j�W 1 for 0Wj < p=3. Usingthe fact that d�0� � 0, we obtain

j4W d�j�Wj for 0Wj <

p3; �3:17�

d�j� is strictly increasing for 0Wj <p3: �3:18�

In the third step of the proof of (3.15) it will be shown that

M�D�j�; o�X �1� const j2�M�a; 6� for 0Wj <p3: �3:19�

Using (3.18), propositions (3.14) and (3.16) may be expressed as follows:

M�D; o�XM�D�j�; o� for 0Wj <p3: �3:20�

M�D�j�; o� is nondecreasing for 0Wj <p3: �3:21�

288 PETER M. GRUBER

De¢ne

k � k�a;j� � r2�a;j� cos2p6� j

� �� 2a cos2

p6� j

� �sin

p2� 2j

� �� 5 sin

p2ÿ 2j

5

� �� ÿ1;

l � l�a;j� � r2�a;j� cos2p6ÿ j

5

� �� 2a cos2

p6ÿ j

5

� �sin

p2� 2j

� �� 5 sin

p2ÿ 2j

5

� �� ÿ1:

Then, considering the de¢nition of D�j� � D�d� in (3.14), one may expressM�D�j�; o� as follows:

M�D�j�; o� � 2Z p

6�j

0

Z k1=2cosc

0g�r2�r dr dc� 10

Z p6ÿj

5

0

Z l1=2cosc

0g�r2�r dr dc

�Z p

6�j

0G

kcos2 c

� �dc� 5

Z p6ÿj

5

0G

lcos2 c

� �dc:

�3:22�

Elementary calculations show that

k�a; 0� � a

2��3p ; kj�a; 0� � ÿ a

3; kjj�a; 0� � ÿ 4a

15��3p ; kj�a;j� < 0;

l�a; 0� � a

2��3p ; lj�a; 0� � ÿ a

15; ljj�a; 0� � 28a

15��3p ; l�a;j�; lj�a;j� > 0;

r2�a;j�X r2�a; 0� � 2a3��3p for 0Wj < 0:2:

�3:23�

Differentiating (3.22), integrating by parts, and taking into account thatÿ2k�a;j�kjj�a;j�; l�a;j�; ljj�a;j� > 0 for 0Wj < 0:2; that g0X 0, andg�0� � 0, we see that

M�D�0�; o� �M�a; 6�;Mj�D�j�; o� � G�r2� � 5

ÿ15

G�r2� �

� kj

Z p6�j

0g

kcos2 c

� �dc

cos2 c� 5lj

Z p6ÿj

5

0g

lcos2 c

� �dc

cos2 c;

Mj�D�0�; o� � 0;


Mjj�D�j�; o� � kjg�r2�

cos2 p6 � jÿ �� 5

ÿ15

ljg�r2�

cos2 p6 ÿ j

5

ÿ �� kjj

Z p6�j

0g

kcos2 c

� �dc

cos2 c� k2j

Z p6�j

0g0

kcos2 c

� �dc

cos4 c�

� 5ljjZ p

6ÿj5

0g

lcos2 c

� �dc

cos2 c� 5l2j

Z p6ÿj

5

0g0

lcos2 c

� �dc

cos4 c

� kjcos2 p

6 � jÿ �ÿ lj

cos2 p6 � j

5

ÿ �� kjj tanp6� j

� ��

� 5ljj tanp6ÿ j

5

� ��g�r2� ÿ 2kkjj

Z p6�j

0g0

kcos2 c

� �sin2 ccos4 c

dc�

� k2j

Z p6�j

0g0

kcos2 c

� �dc

cos4 cÿ 2lljj

Z p6ÿj

5

0g0

lcos2 c

� �sin2 ccos4 c

dc�

� l2j

Z p6ÿj

5

0g0

lcos2 c

� �dc

cos4 c

� aJ�j�g�r2� ÿ aK�j�g�k� � aL�j�g�l� for 0Wj < 0:2; �3:24�

where all the expressions

I�j� � 1k

kjcos2 p

6 � jÿ �ÿ lj

cos2 p6 � j

5

ÿ �� kjj tanp6� j

� �� 5ljj tan

p6ÿ j

5

� � !;

J�j� � I�j� � 1a

k2j2k sin p

6 � jÿ �ÿ ljj tan

p6ÿ j

5

� � !;

K�j� � 1a

k2j2k sin p

6 � jÿ � ; L�j� � 1

aljj tan

p6ÿ j

5

� �do not depend on a. (3.25)

Propositions (3.23) and (3.25) imply that

J�0� � 50��3p ÿ 28225

; K�0� � 50��3p

225; L�0� � 28

225: �3:26�

From 0 < kW lW r2 cos2�p=6ÿ j=5�W 0:286 r2 for 0Wj < 0:2, and (3.3) we obtainthat 0 < g�k�W g�l�W b g�r2� for 0Wj < 0:2, where 0 < b < 1. Hence (3.24) impliesthat

Mjj�D�j�; o�X a�J�j� ÿ �K�j� ÿ L�j�b�g�r2� for 0Wj < 0:2:

Since J�0� ÿ �K�0� ÿ L�0�� 0; K�0� ÿ L�0� > 0, by (3.26), 0 < b < 1, and since

290 PETER M. GRUBER

J�j�; K�j�; L�j� are continuous in j and do not depend on a by (3.25), we concludefurther that

Mjj�D�j�; o�X const a g�r2�X const a g�a�for all sufficiently small a > 0 and for 0WjW const �< 0:2�;

where we have used (3.23) and(3.2). Combining this with (3.24) and (3.8) then yields,

M�D�j�; o�XM�D�0�; o� �Mj�D�0�; o� � 12jj�D�Wj�; o�j2

XM�a; 6� � const a g�a�j2 X �1� consta g�a�M�a; 6�j

2�M�a; 6�

X �1� const j2�M�a; 6�for all sufficiently small a > 0 and for 0WjW const:

Since by (3.21)M�D�j�; o� is nondecreasing for 0Wj < p=3 , we thus obtain (3.19),decreasing the constant in �1� const j2� if necessary.

Finally, noting that D�d� � D�j� for d � d�j�; proposition (3.15) is an immediateconsequence of (3.20) and (3.17).

3.2.5. The form of D.

There is a Landau symbol o�1� as d! 0 for which the following hold: letD andC�d� be as in (3.14) and let pi; i � 1: . . . ; 6; be the mirror images of the center ofC�d� in the lines containing the edges of D. Then there is a regular 6-gon withcenter o, vertices qi, and such that kqik � �2a=

��3p �1=2 and

kpi ÿ qikW o�1�kqik for i � 1; . . . ; 6. (3.27)

This result is an immediate consequence of the next proposition.

Let E > 0 .Then for each su¤ciently small d > 0 one gets: for any convex 6-gonD such that jDj � a and C�d� � C��1ÿ d��a=2 ��

3p �1=2; o� � D there is a regular

convex 6-gon R with center o; jRj � a , and dH �D;R� < E. (3.28)

Here dH is the usual Hausdorff metric on the space of compact convex sets inE2, see[29]. For the proof of (3.28) assume the contrary. Then there are dk > 0 and convex6-gons Dk such that dk ! 0 as k!1; jDkj � a; C�dk� � Dk; and dH �Dk;R�X Efor each regular 6-gon R with center o and jRj � a. SinceC�dk� ! C�0� � C��a=2 ��

3p �1=2; o�; C�dk� � Dk; and jDkj � a; the sequence �Dk�

is bounded. Thus a version of Blaschke`s selection theorem yields the following:by considering a suitable subsequence and renumbering, if necessary, we may assumethat Dk ! D; say, where D is a convex 3, 4, 5, or 6-gon, C�0� � D; jDj � a, and


dH �D;R�X E for each regular 6-gon R with jRj � a and center o. A simpleisoperimetric inequality for polygons says that all convex 3, 4, 5, or 6-gonscircumscribed to C�0� � C��a=2 ��

3p �1=2; o� have area greater than a, except for

the regular 6-gon with center o. ThusDmust be a regular 6-gon with center o. HencedH �D;R� � 0 for the regular 6-gon R � D with jRj � a and center o. This con-tradiction concludes the proof of (3.28) and thus of (3.27).

3.3. PROOF OF THEOREM 2 IN THE EUCLIDEAN CASE

3.3.1. In the ¢rst part we will prove the following proposition.

Let H be a convex 3, 4, 5, or 6-gon in E2 and �Tm� a sequence of ¢nite sets in E2 with#Tm � m for m � 1; 2; . . . ; such that

ZH

minp2Tmff �kpÿ xk�gdx � mM

jHjm; 6

� �as m!1: �3:29�

Then Tm is asymptotically a regular hexagonal pattern of edge length

2jHj��3p

m

� �1=2

:

3.3.2. Simpli¢cations, notations, and remarks. Suitable small distortions of the setsTm do not affect Proposition (3.29). Thus we may assume that no vertex of H isequidistant from 2 points of Tm, no point on an edge of H is equidistant from 3points of Tm, and no point of E2 is equidistant from 4 points of Tm form � 1; 2; . . . We clearly may assume that jHj � 1.

De¢ne the Dirichlet-Voronoi cells

D�Tm; p� � fx 2 H : kxÿ pkW kxÿ qk for each q 2 Tmg; p 2 Tm:

The sets D�Tm; p� are convex polygons in H, possibly degenerate, that is, ofdimension less than 2. For i � 3; 4; . . . ; let Dij; j � 1; . . . ;mi, be the proper, thatis the non-degenerate i-gons among these polygons and for each Dij let pij 2 Tm

be such that Dij � D�Tm; pij�. Put aij � jDijj. Let mprop � m3 � � � � �mk, where kis the maximum i with mi > 0. The following properties are easy to check.

The polygons Dij; i � 3; . . . ; k; j � 1; . . . ;mi, are proper convex polygons whichtile H. A vertex of this tiling in intH is a vertex of precisely 3 such polygons, a

292 PETER M. GRUBER

vertex in the relative interior of an edge of H is a vertex of precisely 2 suchpolygons and a vertex of H is a vertex of precisely one of these polygons. (3.30)

a31 � � � � � a3m3 � � � � � akmk � 1 �� jHj�: �3:31�

The assumptions of proposition (3.29), the fact that f is continuous and strictlyincreasing and the de¢nition of the Dij together with (3.3) imply that

�m� o�m��M 1m; 6

� �X

ZH

minp2Tmff �kpÿ xk�g dx

�Xi;j

ZDij

f �kpij ÿ xk� dx �M�D31; p31� � � � � �M�Dkmk ; pkmk�

XM�a31; 3� � � � � �M�akmk ; k�:

�3:32�

3.3.3. Most polygons D�Tm; p� are proper and contained in intH:

mprop � mÿ o�m�: �3:33�

If (3.33) did not hold, then mprop W sm for in¢nitely many m, where 0 < s < 1. Con-sidering only such m (in the proof of (3.33)), propositions (3.32), (2.10), (1.1), (2.7)and (3.13) together yield a contradiction and thus settle the proof of (3.33):

�m� o�m��M 1m; 6

� �X

ZH


�ZH

minp2Tm�H�

ff �kpÿ xk�g dxXmpropM1

mprop; 6

� �X smM

1sm

; 6� �

> tmM1m; 6

� �; where t > 1:

Next,

mint � #fDij : Dij � intHg � mprop ÿ o�m� � mÿ o�m�: �3:34�

For, if this were not the case, then mint W s2 m for in¢nitely many m, where0 < s < 1. Consider in the following only such m. Let t � w2 > 1 correspond tos as in (3.13). By choosing w �> 1� closer to 1, if necessary, we may suppose thatwW 1=s. Since the ¢rst expression in (3.32) tends to 0 as m!1, (2.9) implies thatmaxfdiamDijg ! 0 as m!1. Choose K � intH homothetic to H withjK j � jHj=w � 1=w. Then for all suf¢ciently large m there are mK Wmint properconvex polygons among the sets Dij \ K . Thus (3.32), (2.10), (1.1) applied to K ,(2.7), (3.13), the inequality wmint W sm and the equality t=w � w > 1 lead to a


contradiction, concluding the proof of (3.34):

�m� o�m��M 1m; 6

� �X

ZH

minp2Tmff �kpÿ xk�g dxX

ZK


X

ZK

minp2Tm�K�

ff �kpÿ xk�g dxXmKMjK jmK

; 6� �

XmintMjK jmint

; 6� �

� 1wwmintM

1wmint

; 6� �

X1wsmM

1sm

; 6� �

XtwmM

1m; 6

� �:

3.3.4. Most Dij are 6-gons. We need the auxiliary results (3.35)^(3.37).

3m3 � � � � � kmk � 6�m3 � � � � �mk� ÿ o�m� � 6mprop ÿ o�m� � 6mÿ o�m�;or 3m3 � � � � � 5m5 � 7m7 � � � � � kmk � 6m 6�6 ÿ o�m� �W 6m�; (3.35)

where m 6�6 � m3 � � � � �m5 �m7 � � � � �mk. Let v; e; n � mprop denote the numbersof vertices, edges, and facets of the tiling of H by the proper convex polygonsDij. Proposition (3.30) shows that

3v � 3m3 � � � � � kmk � vbd � 2vH ; 2e � 3m3 � � � � � kmk � ebd;

where vbd; vH ; ebd are the numbers of vertices of the tiling in the relative interiors ofthe edges of H, of the vertices of H, and of the edges of the tiling on bdH,respectively. Euler's polytope formula then yields that

6 � 6vÿ 6e� 6n � ÿ�3m3 � � � � � kmk� � 6n� 2vbd � 4vH ÿ 3ebd:

Now, noticing that (3.33), (3.34) and vH W 6 imply that n � mprop � mÿ o�m� and2vbd � 4vH ÿ 3ebd ÿ 6 � o�m�, proposition (3.35) follows.

Let D; E be regular polygons with center o and such that D � E and thecircumcircle of D contains the incircle of E and vice versa. Then

14WjDjjEj W 4; �3:36�

as elementary calculations show. The fact that f satis¢es the growth condition andthus is strictly increasing implies the next result.

Let D; E be regular polygons with center o and such that D � E and thecircumcircle of D is contained in the interior of the incircle of E. Then there

294 PETER M. GRUBER

are l > 1 > m > 0 such that

lD � mE; lD 6� int mE; jDj � jEj � jlDj � jmEj;M�lD; o� �M�mE; o� <M�D; o� �M�E; o�: �3:37�

After these preparations we prove that

m6�6 � m3 � � � � �m5 �m7 � � � � �mk � o�m�: �3:38�Assume, on the contrary, that (3.38) does not hold. Then

m6�6 X constm for in¢nitely many m. (3.39)

Only such m will be considered. (3.32), (3.31), (3.36), (3.37) and (3.33) imply that

�m� o�m��M 1m; 6

� �XM�b31; 3� � � � � �M�bkmk ; k�; where

b31 � � � � � bkmk � 1 and15m

W bij W5m

for all sufficiently large m: (3.40)

It follows from (3.35) and (3.39) that the average number i6�6 of vertices of thenon-hexagonal polygons Dij satis¢es

i6�6 � 1m 6�6�3m3 � � � � � 5m5 � 7m7 � � � � � kmk� W 6

! 6; asm!1:�

(3.41)

Let m<6 � m3 � � � � �m5; m>6 � m7 � � � � �mk. The cases m<6 � 0 and m>6 � 0 forin¢nitely many m are excluded by (3.41). Thus we may assume thatm<6; m>6; m 6�6 � m<6 �m>6 > 0 for all suf¢ciently large m. Consider only suchm. Let

b6 � 1m6�b61 � � � � � b6m6 � for m6 > 0 and b6 � 1

motherwise;

b<6 � 1m<6�b31 � � � � � b5m5 �; i<6 �

1m<6�3m3 � � � � � 5m5�;

b>6 � 1m>6�b71 � � � � � bkmk�; i>6 �

1m>6�7m7 � � � � � kmk�;

b 6�6 � 1m 6�6�m<6b<6 �m>6b>6�; i6�6 � 1

m 6�6�m<6i<6 �m>6i>6�: (3.42)

By (3.40),

15m

W b<6; b 6�6; b>6 W5m; i<6 W 5; i>6 X 7: (3.43)

Consider the line segment ��b<6; i<6�; �b>6; i>6�� and parametrize it in the form:�a� bv; v� : i<6 W vW i>6. Clearly,

15m

W a� bvW5m

for 5W vW 7 and a� bi<6 � b<6; a� bi>6 � b>6: (3.44)


This, combined with (3.12) and (3.2), shows that

d2

dv2M�a� bv; v� �Maa�a� bv; v�b2 � 2Mav�a� bv; v�b�Mvv�a� bv; v�

X const1mg

1m

� �for all su¤ciently large m and for 5W vW 7 (3.45)

Next it will be proved that

�1ÿ l�M�a� 5b; 5� � lM�a� 7b; 7�X �1� const �M�a� bv; v�for all su¤ciently large m and for 5:5W v � 5�1ÿ l� � 7lW 6:5. (3.46)

To see this note ¢rst that (3.45) implies the following:

M�a� bi ; i � �M�a� bv; v� � d

dvM�a� bv; v��i ÿ v� �

� 12d2

dv2M�a� b�v� W �i ÿ v�; �v� W �i ÿ v��i ÿ v�2

XM�a� bv; v�� ddv

M�a� bv; v��i ÿ v��const 1mg

1m

� ��i ÿ v�2

for all sufficiently large m and for 5W vW 7 where i � 5 or 7:

This together with 5:5W v � 5�1ÿ l� � 7lW 6:5, (3.44), (3.8) and (3.2) then yields(3.46):

�1ÿ l�M�a� 5b; 5� � lM�a� 7b; 7�

XM�a� bv; v� � 0� const mg1m

� �X �1� const �M�a� bv; v�

for all sufficiently large m and for 5:5W v � 5�1ÿ l� � 7lW 6:5:

Next we prove that

�1ÿ m�M�a� bi<6; i<6� � mM�a� bi>6; i>6�X �1� const �M�a� bv; v�for all su¤ciently large m and for 5:5W v � �1ÿ m�i<6 � mi>6 W 6:5. (3.47)

Given m, choose l such that v � �1ÿ m�i<6 � mi>6 � 5�1ÿ l� � 7l. Since M�� ; �� isconvex by (3.5), propositions (3.46) and (3.44) then yield (3.47).

From (3.40), (3.5), (3.47), (3.43), (3.42), (3.41), (3.5), the fact thatmprop � m6 �m 6�6, m6b6 �m 6�6b 6�6 � 1, (3.33), (3.35) and (3.41) and thus

296 PETER M. GRUBER

m66�m 6�6i 6�6 � 6mprop ÿ o�m�, (3.7), (3.43) and (3.8), we obtain that

�m� o�m��M 1m; 6

� �Xm6M�b6; 6� �m<6M�b<6; i<6� �m>6M�b>6; i>6�

Xm6M�b6; 6� �m 6�6m<6

m 6�6M�b<6; i<6

� ��m>6

m 6�6M�b>6; i>6��

Xm6M�b6; 6� �m 6�6M�b 6�6; i6�6� � const m6�6M�b 6�6; i 6�6�

X �m6 �m 6�6�M 1m6 �m 6�6

�m6b6 �m6�6b 6�6�; 1m6 �m 6�6

�m66�m 6�6i 6�6��

�

� constm 6�6M�b 6�6; i6�6�

X �mÿ o�m��M 1mprop

; 6� �

� constm 6�6M�b 6�6; i6�6�

X �mÿ o�m��M 1m; 6

� �� const m6�6

1mg

1m

� �:

By (3.8), this yields a contradiction to (3.39) and thus concludes the proof of (3.38):

o�m� 1mg

1m

� �Xm 6�6

1mg

1m

� �:

3.3.5. The total area of the non-hexagonal Dij is small. If m 6�6 � 0 for all m with a¢nite set of exceptions, we are ¢nished. Assume now that m6�6 > 0 for in¢nitely manym. Only such m will be considered in the following. Let

a6 � 1m6�a61� � � � � a6m6�; a 6�6 � 1

m 6�6�a31 � � � � � a5m5 � a71 � � � � � akmk�:

Then we have

m6�6a 6�6 � o�1�. (3.48)

The average area of the polygons Dij is 1=mprop and thus by (3.33) is equal to1=�mÿ o�m��. Therefore, if a 6�6 W a6, then a 6�6 W 1=�mÿ o�m�� and (3.46) is animmediate consequence of (3.38). Assume now that a 6�6 X 1=�mÿ o�m��. If (3.48)does not hold, then

m6�6a 6�6 X const �> 0� for in¢nitely many m: (3.49)

Only such m will be considered. Combining (3.32), (3.5), (3.41), (3.7), (3.6) witha � a6 or a 6�6 and b � 1=m, (3.33), (3.31), (3.33), (3.9), (3.10), (3.2) and


a 6�6 X 1=mprop X 1=m, we see that

�m� o�m��M 1m; 6

� �Xm6M�a6; 6� �m 6�6M�a 6�6; i6�6�

Xm6M�a6; 6� �m 6�6M�a 6�6; 6�

X �m6 �m 6�6�M 1m; 6

� �� m6 a6 ÿ 1

m

� ��m6�6 a 6�6 ÿ 1

m

� �� Ma

1m; 6

� ��

�m 6�62

Maa1m� W a 6�6 ÿ 1

m

� �; 6

� �a 6�6 ÿ 1

m

� �2

X �mÿ o�m��M 1m; 6

� �� o�1�g 1

m

� ��

� constm6�6g 1

m� W�a 6�6 ÿ 1m�

ÿ �1m� W�a 6�6 ÿ 1

m�a 6�6 ÿ 1

m

� �2

X �mÿ o�m��M 1m; 6

� �� o�1�g 1

m

� �� const m6�6g

1m

� �a 6�6 ÿ 1

m

ÿ �2a 6�6

:

Thus, taking into account (3.8),

m6�6a 6�6 ÿ 1

m

ÿ �2a 6�6

W o�1�. (3.50)

Since m 6�6 � o�m� by (3.38), inequality (3.49) together with a6�6�X 1=m� shows thata 6�6 is an arbitrarily large multiple of 1=m. Hence the left hand side in (3.50) isessentially equal to m6�6a 6�6. Thus (3.50) and (3.49) are in contradiction and the proofof (3.48) is complete.

3.3.6. The area of most of the polygonsD6j is approximately 1=m. To see this, we ¢rstgive an upper estimate:

1m

W a6j W1m� o

1m

� �for all j with a6j X

1m; (3.51)

with a set of at most o�m� exceptions.

mM�1=m; 6� ! 0; M�0; 6� � 0 and M�a; 6� is strictly increasing and continuous foraX 0. Thus (3.32) implies that a6j all are arbitrarily small for all suf¢ciently largem. From (3.32), (3.6) with a � a6j; b � 1=m, (3.33), (3.38), (3.31), (3.48), (3.33),

298 PETER M. GRUBER

(3.38), (3.10), the remark just made, (3.9) and (3.2) it follows that

�m� o�m��M 1m; 6

� �XM�a61; 6� � � � � �M�a6m6 ; 6�

Xm6M1m; 6

� ��Ma

1m; 6

� �m6a6 ÿm6

m

� ��

� 12

Xm6

j�1Maa

1m� Wj a6j ÿ 1

m

� �; 6

� �a6j ÿ 1

m

� �2

X �mÿ o�m��M 1m; 6

� ��Ma

1m; 6

� ��1ÿ o�1� ÿ �1ÿ o�1��

� constX g 1

m� Wj a6j ÿ 1m

ÿ �ÿ �1m� Wj a6j ÿ 1

m

ÿ � a6j ÿ 1m

� �2

: a6j X1m

( )

X �mÿ o�m��M 1m; 6

� �� o�1�g 1

m

� ��

� const g1m

� �X a6j ÿ 1m

ÿ �2a6j

: a6j X1m

( ):

Thus, using (3.8) and putting a6j � �1� Ej�=m; dj � E2=�1� Ej� for a6j X 1=m andEj � dj � 0 for a6j < 1=m, we conclude that

o�m� 1mg

1m

� �X g

1m

� �Xm6

j�1

E2j�1� Ej�m ; or

Xm6

j�1dj W o�m�: �3:52�

Denote the Landau symbol in (3.52) by �o�m�. Let dj > � �o�m�=m�1=2 for precisely lindices. Then (3.52) implies that l� �o�m�=m�1=2 W �o�m�, or lW o�m�. Hencedj � E2j =�1� Ej�W � �o�m�=m�1=2 � o�1� for all j with a set of at most �lW �o�m�exceptions. Thus, a fortiori, Ej W o�1� for all j with a set of at most o�m� exceptions.Considering the de¢nition of Ej, this concludes the proof of (3.51).

The proof of the corresponding lower estimate

1mÿ o

1m

� �W a6j W

1m

for all j with a6j <1m; (3.53)

with a set of at most o�m� exceptions

is slightly longer. We ¢rst show that

12m

W a6j <1m

for all j with a6j <1m; (3.54)

with a set of at most o�m� exceptions.


Let m6�; m60; m6ÿ; m6ÿÿ be the numbers of indices j with a6j > 1=m;� 1=m; < 1=m,and< 1=2m, respectively. Denote by a6�; . . . ; a6ÿÿ the averages of the correspondingareas. �Ifm � 0, let a � 1=m, say.) Assume now that (3.54) does not hold. Then

�m6ÿ X �m6ÿÿX const m for in¢nitely many m. (3.55)

From now on we consider only suchm. Propositions (3.55), (3.33), (3.38), (3.31) and(3.48) imply that

1mÿ a6ÿX

constm

; a6� ÿ 1m

Xconstm

, (3.56)

m6ÿ �m60 �m6�� m6� � mÿ o�m�; orm6ÿm�m60

m�m6�

m� 1ÿ o�1�;

m6ÿa6ÿ �m60a60 �m6�a6� � 1ÿ o�1�; (3.57)

and thus

m6� a6� ÿ 1m

� �� ÿm6ÿ a6ÿ ÿ 1

m

� �� o�1�. (3.58)

The remark in the proof of (3.51) also shows that a6� is arbitrarily small for allsuf¢ciently large m. Combining (3.32), (3.5), (3.6) with a � a6ÿ; a6� andb � 1=m, (3.57), (3.58), (3.9), (3.10) the remark about a6�, gives:

�m� o�m��M 1m; 6

� �X �m6ÿ �m60 �m6��M 1

m; 6

� �� m6ÿ a6ÿ ÿ 1

m

� ��

�m6� a6� ÿ 1m

� ��Ma

1m; 6

� ��m6�

2Maa

1m� W a6� ÿ 1

m

� �;6

� �a6�ÿ 1

m

� �2

X �mÿ o�m��M 1m; 6

� �� o�1�g 1

m

� �� const m6�g

1m

� � �a6� ÿ 1m�2

a6�:

Hence, using (3.8), (3.58) and (3.55), resp. (3.56), it follows that

o�m� 1mg

1m

� �X o�1�g 1

m

� ��m6 � a6� ÿ 1

m

� �a6� ÿ 1

m

a6�g

1m

� �;

or

o�1�X m6 ÿ 1mÿ a6ÿ

� �� o�1�

� �a6� ÿ 1

m

a6�or o�1�X a6� ÿ 1

m

a6�:

If a6�X 2=m we would have o�1�X 1=2. If a6�W 2=m, (3.56) would implyo�1�X const. Both alternatives are impossible for suf¢ciently large m, and sothe proof of (3.54) is complete.

300 PETER M. GRUBER

In view of (3.54) it is suf¢cient for the proof of (3.53) to consider those j with1=2mW a6j < 1=m. In this case the proof of (3.53) is similar to that of (3.51) andis therefore omitted.

3.3.7. Before proceeding to the ¢nal steps of the proof of proposition (3.29), wecombine the results (3.33), (3.38), (3.34), (3.51), (3.53) and (3.48):

Among the mÿ o�m� proper convex polygons Dij all up to o�m� are 6-gons con-tained in intH of area beween 1=mÿ o�1=m� and 1=m� o�1=m�. The total areaof these 6-gons D6j is 1ÿ o�1�. Call these D6j and also the corresponding pointsp6j nice. (3.59)

Next we show that

each of the mÿ o�m� nice points p6j 2 Tm, with a set of at most o�m� exceptions,has the following property: the 6 points p1; . . . ; p6 2 Tm which together withp6j determine D6j (i.e. D6j � fx : kxÿ p6jkW kxÿ p1k; . . . ; kxÿ p6kg� formup to o�1� a regular 6-gon with center p6j and edge length �2= ��

3p

m�1=2. A nicepoint with this property will be called good and the corresponding pointsp1; . . . ; p6 its neighbors. A point in Tm which is not good is bad. (3.60)

For each nice D6j let C6j � C��1ÿ dj��a6j=2��3p �1=2; p6j� be the circle with center p6j

and maximum radius contained in D6j . Then (3.32), (3.15), (3.6) with a � a6jand b � 1=m, and (3.59) imply the following:

�m� o�m��M 1m; 6

� �XXm6

j�1M�D6j; p6j�X

Xm6

j�1�1� const d2j �M�a6j; 6�

XX�1� const d2j �M

1m; 6

� ��Ma

1m; 6

� �a6j ÿ 1

m

� �: D6j nice

� �X �mÿ o�m��M 1

m; 6

� �� const M

1m; 6

� �Xfd2j : D6j niceg ÿ

ÿ �mÿ o�m��Ma1m; 6

� �o

1m

� �;

or

o�m�M 1m; 6

� �� o�1�Ma

1m; 6

� �XM

1m; 6

� �Xfd2j : D6j niceg;


or

Xfd2j : D6j nicegW o�m�

by (3.8) and (3.9). Since by (3.59) there are mÿ o�m� nice D6j, we have to considermÿ o�m� numbers d2j X 0 with sum at most o�m�. An argument similar to theone in the proof of (3.51) then shows that d2j W o�1� for all d2j with a set of at mosto�m� exceptions. This together with (3.28) ¢nally yields (3.60).

All points of Tm, with a set of at most o�m� exceptions, are very good, that is, theyare good and their neighbors are also good. (3.61)

By (3.60) the distance of a good point to any of its neighbors is �1� �o�1��2= ��3p

m�1=2where �o�1� is the Landau symbol o�1� in (3.60). For each bad point q consider allgood points contained in the circle C��1� �o�1��2= ��

3p

m�1=2; q�. For each of thesegood points p consider the circle C��1ÿ �o�1��1=2 ��

3p

m�1=2; p� � D�Tm; p�.These circles do not overlap and all are contained in the circleC��1� �o�1��3=2��2= ��

3p

m�1=2; q�. Comparing areas, we see that there are at most9 such good circles for suf¢ciently small �o�1�, that is, for suf¢ciently large m.Now cancel all bad points and for each bad point q cancel the (at most 9) goodpoints in C��1� �o�1��2= ��

3p

m�1=2; q�. Since by (3.60) there are at most o�m� badpoints, this amounts to the cancellation of at most o�m� points. Clearly, each ofthe remaining mÿ o�m� ÿ o�m� � mÿ o�m� good points has no bad neighborand thus is very good, concluding the proof of (3.61).

Finally,

if p is a very good point and p1; . . . ; p6 are its neighbors,

then C�1:15�2= ��3p

m�1=2; p� \ Tm � fp; p1; . . . ; p6g: (3.62)

To see this note that by (3.61) p; p1; . . . ; p6 are contained in the circle in (3.62) andthis circle is contained in D�Tm; p� [D�Tm; p1� [ . . . [D�Tm; p6� as an elementaryargument shows. This yields (3.62).

3.3.8. Proposition (3.29) now follows from (3.60)-(3.62). &

3.4. PROOF OF THEOREM 2 IN THE RIEMANNIAN CASE

3.4.1. The ¢rst part consists in showing that

Sn is uniformly distributed in J. (3.63)

302 PETER M. GRUBER

The function f is continuous with f �0� � 0.We use the growth condition to see thatthe right side in (3.1) tends to 0 as n!1. HenceZ

Jminp2Sn

ff �RM�p; x��g doM�x� ! 0 �3:64�

In the ¢rst step of the proof of (3.63) we establish the following proposition, wherefor the de¢nitions of Sn�K� and n�K� the reader is referred to (2.10):

Let K � J be Jordan measurable with 0 < oM�K� < oM�J�. ThenZK

minp2Sn�K�

ff �RM�p; x��g doM�x� � n�K�M oM�K�n�K� ; 6

� �: �3:65�

Since K is Jordan measurable and oM�K� > 0, we have intK 6� ;. By (2.10) and(3.64) the integral in (3.65) tends to 0 as n!1. An application of (2.9) to K insteadof J then shows that n�K� � #Sn�K� ! 1 as n!1. Thus if (3.65) did not hold,Theorem 1 applied to K implies thatZ

Kmin

p2Sn�K�ff �RM�p; x��g doM�x�X l n�K�M oM�K�

n�K� ; 6� �

for infinitely many n; where l > 1: �3:66�

From now on only such nwill be considered (in the proof of (3.65)). By (2.8) there is am > 1 such that lM�a; 6�XM�m a; 6� for all suf¢ciently small aX 0. Choose acompact Jordan measurable set L � int�J n K� with

oM�L� > 0 and moM�K� � oM�L�X noM�J�, where n > 1. (3.67)

Since our choice of L implies that the sets K and L have positive distance, we con-clude from (2.9) and (2.10) that

K \ L � ;; Sn�K� \ Sn�L� � ;; n�K [ L� � n�K� � n�L�W n

(3.68)for all su¤ciently large n.

Using the argument that was applied to K we get n�L� ! 1. Hence Theorem 1applied to L shows thatZ

Lmin

p2Sn�L�ff �RM�p; x��g doM�x�>� n�L�M oM�L�

n�L� ; 6� �

: �3:69�

(The symbol >� means the following: the limit inferior of the quotient of the left andthe right side in (3.69) as n!1 is at least 1.) By (2.8) we may choose x > 1 such thatM�n a; 6�X xM�a; 6� for all suf¢ciently small aX 0. Then, using obviousabbreviations, (2.1), (3.1), (3.68) (2.10), (3.66), (3.69), our choice of m, (3.5) and


Jensen's inequality, (3.68), (2.7), (2.6), (3.67) and our choice of x yield a con-tradiction and thus conclude the proof of (3.65):

nMoM�J�

n; 6

� ��ZJ

X

ZK�ZL

>� l n�K�M oM�K�n�K� ; 6

� �� n�L�M oM�L�

n�L� ; 6� �

X n�K�M moM�K�n�K� ; 6

� �� n�L�M oM�L�

n�L� ; 6� �

X �n�K� � n�L��M moM�K� � oM�L�n�K� � n�L� ; 6

� �X nM

noM�J�n

; 6� �

X x nMoM�J�

n; 6

� �for all sufficiently large n:

In the second step we show that

n�J� � n. (3.70)

Otherwise n�J� < sn for in¢nitely many n, where 0 < s < 1. Only such n will be con-sidered. By (3.13) there is t > 1 such that s nM�a=s n; 6�X t nM�a=n; 6� for eachaX 0 and all suf¢ciently large n. Then (2.1), (3.1), (2.10) with K � J, Theorem1, and (2.7) yield a contradiction, concluding the proof of (3.70):

nMoM�J�

n; 6

� ��ZJ� n�J�M oM�J�

n�J� ; 6� �

X s nMoM�J�s n

; 6� �

X t nMoM�J�

n; 6

� �for all sufficiently large n:

We come to the third step:

Let K � J be Jordan measurable. Then, (3.71)

(i) n�K� � o�n� for oM�K� � 0,(ii) n�K� � n for oM�K� � oM�J�,(iii) n�K� � oM�K�

oM�J� n for 0 < oM�K� < oM�J�.

Assume ¢rst that oM�K� � 0. If (3.71 i) did not hold, n�K�X �Zÿ 1�n for in¢nitelymany n, where Z > 1. Only such n will be considered. Choose a compact Jordan

304 PETER M. GRUBER

measurable set L � int �J n K�withoM�L� > 0. SinceK and L have positive distance,(2.9) and (2.10) imply that

Sn�K� \ Sn�L� � ;; n�K [ L� � n�K� � n�L�X Z n�L�for all su¤ciently large n.

As before, n�L� ! 1. By (3.13) we may choose 0 < z < 1 such thatZmM�a=Zm; 6�W zmM�a=m; 6� for each aX 0 and all suf¢ciently large m. Theseremarks together with (3.65) and (2.7) imply the following contradiction and thus¢nish the proof of (3.71 i):

n�L�M oM�L�n�L� ; 6

� ��ZL�ZK[L� n�K [ L�M oM�K [ L�

n�K [ L� ; 6� �

W Z n�L�M oM�L�Z n�L� ; 6

� �W z n�L�M oM�L�

n�L� ; 6� �

for all sufficiently large n:

Next assume that oM�K� � oM�J�. Then J � �J n K� [ K where oM�J n K� � 0.Hence n � n�J�W n�J n K� � n�K� � o�n� � n�K�W o�n� � n by (3.70) and (3.71 i).Thus n�K� � n and the proof of (3.71 ii) is complete. Finally assume that0 < oM�K� < oM�J�. Then L � J n K also satis¢es 0 < oM�L� < oM�J�. Clearly,

n � n�J�W n�K� � n�L� W n�J� � n�bdK� � n�bdL�W n� o�n�;X n�J� ÿ n�bdK� ÿ n�bdL�X nÿ o�n�:

(�3:72�

by (3.70), (3.71 i), (2.10) and since the Dirichlet-Voronoi cellsfx 2M : RM�p; x�W RM�q; x� for all q 2 Sng; p 2 Sn, all are connected. If (3.71 iii)were not true, then (3.70) shows that either

n�K�W �1ÿ k�oM�K�oM�J� n

and thusoM�K�n�K� ÿ

oM�J�n

X koM�K�n�K�

for in¢nitely many n, where 0 < k < 1, (3.73)

or the corresponding inequality with K replaced by Lwould be true. In the latter caseexchange K and L in the following argument. We consider only n satisfying (3.73).


From (3.6), (3.9), (3.10), (3.2), and (3.73) we obtain that

MoM�K�n�K� ; 6

� �

�MoM�J�

n; 6

� ��Ma

oM�J�n

; 6� �

oM�K�n�K� ÿ

oM�J�n

� ��

� 12Maa

oM�J�n� W


oM�J�n

� �; 6

� �oM�K�n�K� ÿ

oM�J�n

� �2

XMoM�J�

n; 6

� ��Ma

oM�J�n

; 6� �


oM�J�n

� ��

� constgÿoM �J�

n

�oM �K�n�K�


oM�J�n

� �2

;

MoM�L�n�L� ; 6

� �XM

oM�J�n

; 6� �

�MaoM�J�

n; 6

� �oM�L�n�L� ÿ

oM�J�n

� �for all sufficiently large n. �3:74�

Then (2.1), (3.1), J � K [ L; K \ L � ;, (2.10), (3.65), (3.74), (3.73), (3.72), (3.8),(3.9) and (3.2) yield a contradiction:

MoM�J�

n; 6

� �� 1

n

ZJ

� 1n

ZK� 1n

ZL� n�K�

nM

oM�K�n�K� ; 6

� �� n�L�

nM

oM�L�n�L� ; 6

� �

Xn�K� � n�L�

nM

oM�J�n

; 6� �

�

�MaoM�J�

n; 6

� �oM�K� � oM�L�

nÿ n�K� � n�L�

n2oM�J�

� ��

� const goM�K�

n

� �n�K�

oM�K�oM�K�2n�K�2

Xnÿ o�n�

nM

oM�J�n

; 6� �

�MaoM�J�

n; 6

� �1nÿ n� o�n�

n2

� �oM�J� �

� const goM�K�

n

� �oM�K�

n;

or

o�1�g oM�J�n

� �oM�J�

nX g

oM�J�n

� �o

1n

� �� g

oM�J�n

� �oM�K�

n;

306 PETER M. GRUBER

or o�1�X const > 0 for all suf¢ciently large n. This concludes the proof of (3.71 iii)and thus of (3.71).

Now, in order to prove (3.63), we note that for any Jordan measurable set K � Jwith oM�K� > 0 Proposition (3.71) implies that

#�K \ Sn�W#Sn�K� � n�K�X#Sn�K� ÿ #Sn�bdK� � n�K� ÿ n�bdK�

( )� oM�K�

oM�J� n:

3.4.2. In the second part we show the following proposition:

Let 0 < EW 0:1. Then for all su¤ciently large n we have: each p 2 Sn, with aset of at most En exceptions, is, up to E, the center of a regular 6-gon withvertices p1; . . . ; p6 2 Sn of edge length Rn � �2oM�J�=

��3p

n�1=2 andfx : RM�p; x�W 1:1 Rng \ Sn � fp; p1; . . . ; p6g. (3.75)

In the following all distances in E2 will be with respect to the norm q1=2� : �, respect-ively q1=2l � : �, see below.

First, some preparations are needed. In the proof of (3.75) the following conse-quence of Proposition (3.29) (taking into account (3.62)) will be used.

Let 0 < E < 1. Then for all su¤ciently small d > 0 we have: let q� : � be a positivede¢nite quadratic form on E2 and Q0 � E2 a unit square. Then each ¢niteset S0 � E2 for which m � #S0 is su¤ciently large and for whichZ 0

Qminr2S0ff �q1=2�rÿ s��gds�det q�1=2 W �1� d�mM

jQ0j�det q�1=2m

; 6

!;

has the following property: each r 2 S0, with a set of at most �E=3�m exceptions, isup to E=3 the center of a regular 6-gon with vertices r1; . . . ; r6 2 S0, edge lengthR � �2jQ0j�det q�1=2= ��

3p

m�1=2 and such that fs : q1=2�rÿ s�W 1:15Rg \ S0 �fr; r1; . . . ; r6g. (3.76)

Let 0 < EW 0:1 and choose a corresponding d > 0 according to (3.76). Let l > 1 andn > 1 correspond to it by the growth condition for f and M (see (2.8)), both so closeto 1 that

f �l t�W n f �t� for all su¤ciently small tX 0,M�l a; 6�W nM�a; 6� for all su¤ciently small aX 0,

l2n2 W 1� d; 1:1lW1:15l; 1� EW 1:1,

1� E3

� �lÿ 1ÿ E

3

� � 1l

� �1� E

6

� �l� 1� E

6

� �lÿ 1

lW E. (3.77)


For this l > 1 choose pl 2 J; Ul; Vl; hl � '' 0 ''; ql� : �; Il , sets Qli � int Il � Ul;

Snli � Sn�Qli�; nli � #Snli; l � 1; . . . ; k; i � 1; . . . ; il , as in 2.4 such that in particular(2.11)-(2.14), (2.17 i,ii), (2.19 i,ii) hold, and, in addition, we have

oM�J n[l;i

Qli� < E3oM�J�, (3.78)

nli � oM�Qli�oM�J� n, (3.79)

where (3.79) is an immediate consequence of (3.71).Second,

for all su¤ciently large n hold

ZQ0li

minr2S0nliff �q1=2l �rÿ s��gds�det ql�1=2 W �1� d�nliM jQ0lij�det ql�1=2

nli; 6

� �: �3:80�

We use (2.17 i) and (2.19 i) to see that Qli;Snli � Ul . Combining this with the de¢-nition of the integral in M, (2.12), (2.11), the fact that f is nondecreasing, (3.77),(3.65), (2.13), (3.7) and (3.77) shows that, for all suf¢ciently large n,Z

Q0li

minr2S0nliff �q1=2l �rÿ s��g ds �det ql�1=2 W l

ZQli

minp2Snli

ff �l RM�p; x��g doM�x�

W l2n nliMoM�Qli�

nli; 6

� �W l2n nliM

l jQ0lij�det ql�1=2nli

; 6

!

W l2n2 nliMjQ0lij�det ql�1=2

nli; 6

!W �1� d�nliM jQ0lij�det ql�1=2

nli; 6

!.

This completes the proof of (3.80).Third,

for all su¤ciently large n hold: for all pairs l; i, each of the nli points r 2 S0nli, witha set of at most �2E=3�nli exceptions, is up to E=3 the center of a regular 6-gon withvertices r1; . . . ; r6 2 S0nli of edge length R0nli where

2jQlij�det ql�1=2��3p

nli

!1=2

W R0nli W 1� E6

� � 2jQ0lij�det ql�1=2��3p

nli

!1=2

;

fs : q1=2l �rÿ s�W 1:15R0nlig � Q0li;

fs : q1=2l �rÿ s�W 1:15R0nlig \ S0nli � fr; r1; . . . ; r6g: �3:81�

308 PETER M. GRUBER

The square Q0li has edge length 1=kl , see (2.17 ii). Let R0li � Q0li be a parallel strip ofwidth 2W along bdQ0li, where W is so small that for Rli � hÿ1l �R0li� we haveoM�Rli�=oM�Qli� < E=6. Then (3.71 iii) applied to Qli and Rli implies that

n�Rli� < E6nli for all su¤ciently large n. (3.82)

By (2.9) and (2.10),

each point of S0nli has distance < W from Q0li for all su¤ciently large n. (3.83)

Let Q0 be a square of edge length 1 which is obtained by ¢tting together edge-to-edgeQ0li and suitable k2 ÿ 1 translates of it. Let S0 be the union of S0nli and its correspond-ing translates, and similarly for S�R�0 and R0. Since S0nli n Sn�Rli�0 � intQ0li, (3.82)yields,

1ÿ E6

� �k2l nli Wm � #S0W k2l nli and #S�R�0W E

6k2l nli

for all su¤ciently large n. (3.84)

The de¢nition of Q0 and S0 and propositions (3.80), (3.84) and (2.7) together implythat Z

Q0minr2S0ff �q1=2l �rÿ s��gds �det ql�1=2 W k2l

ZQ0li

minr2S0nliff �q1=2l �rÿ s��g ds �det ql�1=2

W �1� d�k2l nliMk2l jQ0lij�det ql�1=2

k2l nli; 6

!W �1� d�mM

jQ0j�det ql�1=2m

; 6

!for all sufficiently large n:

This permits us to apply (3.76). Thus, using (3.84) and the inequality�1ÿ E=6�ÿ1=2 W 1� E=6, we see that

for all su¤ciently large n: each of the m points r 2 S0, with a set of at most �E=3�mexceptions, is up to E=3 the center of a regular 6-gon with vertices r1; . . . ; r6 2 S0,edge length R0nli � �2jQ0j�det ql�1=2=

��3p

m�1=2 and such that fs : q1=2l �rÿ s�W1:15R0nlig \ S0 � fr; r1; . . . ; r6g. For R0nli the following inequalities hold:

2jQ0lij�det ql�1=2��3p

nli

!1=2

W R0nli W 1� E6


nli

!1=2

;

1:15R0nli�1� E=3� < W: �3:85�

From the mX �1ÿ E=6�k2l nli points of S0 cancel ¢rst the at most�2E=6�m�W �2E=6�k2l nli� exceptional points and, second, the at most �E=6�k2l nli points


of S0�R0�, compare (3.84). There remain at least �1ÿ 2E=3�k2l nli points which consistof k2l families. The ¢rst such family is contained in Q0li n R0li. It has at least�1ÿ 2E=3�nli points r, each of which has the properties described in (3.85) and since1:15R0nli�1� E=3� < W also the 6 neighbors of r are contained in Q0li and by (3.83) alsoin S0nli. The proof of (3.81) is complete.

Applying hÿ1l we will show, fourth, that (3.81) implies the following:

for all su¤ciently large n: for all pairs l; i each of the nli points p 2 Snli, with a setof at most �2E=3�nli exceptions, is up to E the center of a regular 6-gonwith vertices p1; . . . ; p6 2 Snli of edge length Rn � �2oM�J�=

��3p

n�1=2 andfx : RM�p; x�W 1:1Rng \ Sn � fp; p1; . . . ; p6g. (3.86)

Let p 2 Snli be such that r � p0 2 S0nli is one of the at least �1ÿ 2E=3�nli points asdescribed in (3.81). Let r1; . . . ; r6 2 S0nli be its neighbors and let pj � hÿ1l �rj�. Then(3.81), (2.11), and (3.77) imply that

for all su¤ciently large n:

(i) 1ÿ E3

� � 1lR0nli W

1lq1=2l �rÿ rj�

W RM�p; pj�W lq1=2l �rÿ rj�W 1� E3

� �lR0nli W 1:1lR0nli,

and similarly for RM�pj; pj�1�,

(ii) fx : RM�p; x�W 1:1lR0nlig � fx : RM�p; x�W1:15l

R0nlig� hÿ1�fs : q1=2l �rÿ s�W 1:15R0nlig� � Qli,

(iii) fx : RM�p; x�W 1:1lR0nlig \ Sn � fp; p1; . . . ; p6g. (3.87)

We estimate Rn, de¢ned in (3.86). From (3.85), (2.13) and (3.79) it follows that

R0nli

W 1� E6


nli

!1=2

W 1� E6

� �l1=2

2oM�Qli��3p

nli

� �1=2

X2jQ0lij�det ql�1=2��

3p

nli

!1=2

X lÿ1=22oM�Qli��

3p

nli

� �1=2

8>>>>><>>>>>:W 1� E

6

� �lRn;

X lÿ1Rn;

8><>:or,

1lRn W R0nli W 1� E

6

� �lRn for all su¤ciently large n. (3.88)

310 PETER M. GRUBER

Thus we can conclude from (3.87 i) and (3.77) that

jRM�p; pj� ÿ RnjW jRM�p; pj� ÿ R0nlij � jR0nli ÿ Rnj

W 1� E3

� �lÿ 1ÿ E

3

� � 1l

� �R0nli � 1� E

6

� �lÿ 1

l

� �Rn

W 1� E3

� �lÿ 1ÿ E

3

� � 1l

� �1� E

6

� �l� 1� E

6

� �lÿ 1

l

� �Rn W ERn;

jRM�pj; pj�1� ÿ RnjW . . . W ERn

for all su¤ciently large n. (3.89)

Hence, up to E, p is the center of a regular 6-gon in Sn with vertices p1; . . . ; p6 2 Sn ofedge length Rn. Finally, from (3.87 ii,iii), (3.88) and (3.89) we obtainfx : RM�p; x�W 1:1Rng \ Sn � fp; p1; . . . ; p6g. The proof of (3.86) is complete.

Fifth, we show that

for all su¤ciently large n each point p 2 Sn, with a set of at most En exceptions,has the properties described in (3.86). (3.90)

By (3.78)and (3.71) n�J nSQli� < �E=3�n and by (2.19 ii) n11 � � � � � nkik W n for allsuf¢ciently large n. Cancel from Sn all points of Sn�J n

SQli� and for each pair

l; i the �2E=3�nli points which do not have the properties described in (3.86). Theremaining �1ÿ E�n points of Sn clearly have the properties described in (3.86).

(3.86) and (3.90) together yield (3.75).

3.4.3. We ¢nally show that (3.75) implies that

Sn is asymptotically a regular hexagonal pattern of edge length2oM�J��

3p

n

� �1=2

(3.93)

Let Ek � 1=k for k � 10; 11; . . ., and choose n10 < n11 < . . . such that (3.75) withE � Ek holds for nX nk. De¢ne g�n� � n; h�n� � 1 for 1W n < n10 andg�n� � n=k; h�n� � 1=k for nk W n < nk�1. Then (3.75) says that each p 2 Sn, witha set of at most g�n� exceptions, is up to h�n� the center of a regular 6-gon withvertices p1; . . . ; p6 2 Sn of edge length �2oM�J�=

��3p

n�1=2 and fx : RM�p; x�W 1:1Rng\Sn � fp; p1; . . . ; p6g. Since g�n� � o�n� and h�n� � o�1�, the proof of (3.91) is com-plete.

3.4.4. Propositions (3.63) and (3.91) together yield Theorem 2. &


4. Volume Approximation of Convex Bodies

4.1. THE RESULT

If the boundary of a convex body C of class C2 with Gauss curvature kC > 0 isendowed with a Riemannian metric, this metric induces, in any tangent plane ofbdC, an associated Euclidean metric. When considering tangent planes, we alwaysmean this metric. For the Riemannian metric of the ¢rst fundamental form ofequi-af¢ne differential geometry see [26, 28]. The ordinary surface area measureon bdC is denoted by o.

THEOREM 3. Let C be a convex body in E3 of class C2 with kC > 0. Let bdC beendowed with the Riemannian metric of the ¢rst fundamental form of equi-af¢nedifferential geometry. Then the following statements hold:

(i) dV �C;Pc�n��

5o2EA�bdC�36

��3p

nas n!1; where oEA�bdC� �

ZbdC

k1=4C �x�do�x�:

(ii) Let the convex polytope Pn 2 Pc�n� be best approximating of C with respect to the

symmetric di¡erence metric dV for n � 4; 5; . . .. Then, as n!1, the setsC \ bdPn are uniformly distributed in bdC and the facets F of Pn areasymptotically regular hexagons with centers at the points of C \ F and edge length

2oEA�bd C�3��3p

n

� �1=2

:

(i) is simply a restatement of the result in Gruber [14]. (i), as well as the remark aboutthe uniform distribution in (ii), extends to all dimensions, see Gruber [15] andGlasauer and Gruber [13]. The result (ii) complements analogous results of Gruber[18] for the Hausdorff metric, the Banach-Mazur distance and a measure of distancedue to Schneider. A result corresponding to Theorem 3 for the mean width approxi-mation of C by inscribed convex polytopes can be proved along the lines of theproof of Theorem 3. Karoly Bo« ro« czky, Jr. has informed us of a result similar toTheorem 3 for general polytopes instead of circumscribed ones.

4.2. THE PROOF

4.2.1. We consider two different ways in which bdC can be thought of as aRiemannian 2-manifold. First we can make use of the ordinary second differentialform and, for the second, we use the ¢rst differential form of equi-af¢ne differentialgeometry. For the latter see [26, 28]. Let RII ; REA;oII ;oEA denote the correspondingRiemannian metrics and area measures on bdC. If J � bdC is Jordan measurablewith respect to o, then it is also Jordan measurable with respect to oEA and viceversa.

312 PETER M. GRUBER

The following propositions are either well known or easy to show using thede¢nitions; k:k is the ordinary Euclidean norm.

doII �x� � k1=4C �x�doEA�x� for x 2 bdC. (4.1)

For each l > 1 there is d > 0 such that

1lW

RII �x; y�REA�x; y�k1=8C �x�

W l for x; y 2 bdC; kxÿ ykW d. (4.2)


12l

R2II �x; y�W dist�x;HC�y��W l2R2II �x; y� for x; y 2 bdC; kxÿ ykW d;

where dist�x;HC�y�� is the distance between x and the support planeHC�y� of Cat y, measured along the normal of bdC through x. (4.3)


1l

ZU

dist�x;HC�x��do�x�WV �U;HC�y��W lZU

dist�x;HC�y��do�x�

for all y 2 bdC and all Jordan measurable neighborhoods U of y of diameterW d, where by V �U;HC�y�� we mean the volume of the set of all points ona normal through a point of U between U and HC�y�. (4.4)

4.2.2. For n � 4; 5; . . ., choose Sn � bdC with #Sn � n such that

12

ZbdC

minp2Sn

fR2EA�p; x�gdoEA�x� � n2

ZHn

kxk2dx � 5o2EA�bdC�36

��3p

nas n!1: (4.5)

Since the right side in (4.5) tends to 0, the sets Sn become increasingly dense in bdCif n is suf¢ciently large. Thus the intersection of the supporting halfspaces of C at thepoints of Sn is a convex polytope Qn, say, for all suf¢ciently large n. The maximumdiameter of the facets of Qn tends to 0. Thus (4.5), (4.1)-(4.4) and the fact thatPn is best approximating imply that

5o2EA�bdC�36

��3p

n� 1

2

ZbdC

minp2Sn

fR2EA�p; x�gdoEA�x� �

� 12

ZbdC

minp2Sn

fR2EA�p; x�gk1=4C do�x� � 12

ZbdC

minp2Sn

fR2II �p; x�gdo�x� �

� dV �C;Qn�X dV �C;Pn�� as n!1: (4.6)


Since dV �C;Pn� ! 0, the maximum diameter of the facets of Pn tends to 0. Thus(4.4)-(4.1) and Theorem 1 imply that

dV �C;Pn�� 12

ZbdC

minp2C\bdPn

fR2II �p; x�gdo�x� �

� 12

ZbdC

minp2C\bdPn

fR2EA�p; x�gk1=4C do�x�

� 12

ZbdC

minp2C\bdPn

fR2EA�p; x�gdoEA�x�>�5o2

EA�bdC�36

��3p

nas n!1: (4.7)

Propositions (4.6) and (4.7) together imply Theorem 3(i) and, in addition,

12

ZbdC

minp2C\bdPn

fR2EA�p; x�gdoEA�x� � 5o2EA�bdC�36

��3p

nas n!1:

Theorem 2 then shows that C \ bdPn is uniformly distributed in bdC andasymptotically forms a regular hexagonal pattern in bdC of edge length equalto �2oEA�bdC�=3 ��

3p

n�1=2 with respect to the Riemannian metric of the ¢rst funda-mental form of equi-af¢ne differential geometry. Elementary arguments now implythat each p 2 C \ bdPn, with a set of at most o�n� exceptions, is up to o�1� the centerof a regular hexagonal facet of Pn of edge length �2oEA�bdC�= ��

3p

n�1=2 with respectto the associated Euclidean metric in the plane containing the facet. This concludesthe proof of Theorem 3(ii). &

5. The Isoperimetric Problem for Polytopes in Minkowski Spaces

5.1. THE RESULT

THEOREM 4. Let I be an isoperimetrix in E3 of class C2 with positive Gausscurvature. Let bd I be endowed with the Riemannian metric of the ¢rst fundamentalform of equi-af¢ne differential geometry. For n � 4; 5; . . ., let Pn be a convex polytopein E3 with n facets and minimum isoperimetric quotient SI �Pn�3=V �Pn�2. Then thefollowing are true:

(i)SI �Pn�3V �Pn�2

� 27V �I� � 15o2EA�bd I�4��3p

nas n!1:

(ii) By replacing Pn by a homothetic copy, if necessary, we may assume that Pn iscircumscribed to I for n � 4; 5; . . .Then, as n!1, the set I \ bdPn is uniformlydistributed in bd I and the facets F of Pn are asymptotically regular hexagons with

314 PETER M. GRUBER

center I \ F and edge length

2oEA�bd I�3��3p

n

� �1=2

:

(i) and the statement in (ii) about the uniform distribution of I \ Pn in bd I extend toall dimensions. This can be proved using results of Gruber [15] and Glasauer andGruber [13].

5.2. THE PROOF

Diskant's result [5] allows us to assume that Pn is circumscribed to I . Using thede¢nitions of SI � : � and V � : �, we have, as in the Euclidean case,SI �Pn� � 3V �Pn�. Thus

SI �Pn�3V �Pn�2

� 27V �Pn�: �5:1�

Hence, since Pn minimizes the isoperimetric quotient, it must have minimum volumeamong all convex polytopes with at most n facets and which are circumscribed to I .Equality (5.1) together with Theorem 3 thus yields Theorem 4. &

6. Optimal Numerical Integration Formulas

In this section we assume that M is of class C3 with metric tensor¢eld of class C2,compare Section 2.

6.1. THE RESULT

A modulus of continuity is a function f : �0;1� ! �0;1� such thatf �s� t�W f �s� � f �t� for all s; tX 0. Recall the de¢nitions of nodes, weights, errorand minimum error from Section 1.

THEOREM 5. Let f : �0;1� ! �0;1� be a modulus of continuity which satis¢es thegrowth condition, let J be a Jordan measurable set in M with oM�J� > 0 and letHf � fg : J ! R : jg�x� ÿ g�y�jW f �RM�x; y�� for all x; y 2 Jg. Then the followingstatements hold:

(i) E�Hf ; n� � infNn;WnfE�Hf ;Nn;Wn�g � n

ZHn

f �kxk�dx as n!1:

(ii) Let �Nn� and �Wn� be sequences of nodes and weights such that

E�Hf ;Nn;Wn� � E�Hf ; n� as n!1:


Then Nn is uniformly distributed and forms asymptotically a regular hexagonalpattern in J of edge length

2oM�J��3p

n

� �1=2

:

Note that from the proof of Theorem 1 it follows that, in principle, it is possible toconstruct sequences �Nn� and �Wn� of nodes and weights such thatE�Hf ;Nn;Wn� � E�Hf ; n�. The statement about the uniform distribution in (ii)can be extended to all dimensions.

6.2. THE PROOF

In essence, our proof of part (i) follows Babenko [1] but with additional dif¢culties.

6.2.1. First some preparations:

For each point in M there is a (geodesically) convex open neighborhood N suchthat the following hold: for any p; q 2 N; p 6� q, the bisectorBN �p; q� � fx 2 N : RM�p; x� � RM�q; x�g is Jordan measurable of measure 0.

(6.1)

SinceM is of class C3 with metric tensor¢eld of class C2, each point ofM has an openneighborhood N in M with the following properties: (i) N is contained in a Jordanmeasurable neighborhood. (ii) Any two points x; y 2 N are connected by a uniquegeodesic segment of length RM�x; y� in N; thus, in particular, N is convex. (iii)For each p 2 N the geodesics starting at p cover N n fpg schlicht. (iv) For eachp 2 N there is a diffeomorphism Dp : N n fpg ! �0; 2p� � �0;�1� such thatDp�N n fpg� � �0; 2p� � �0; a� for a suitable a > 0 and for each geodesic G startingat p, the image Dp�G \ �N n fpg�� is a vertical line segment of the formfbg � �0; c�. See Brauner [4] and Kobayashi and Nomizu [25].

Now, let p; q 2 N; p 6� q. By (ii), each geodesic starting at p contains at most onepoint of the bisector BN �p; q� and the bisector obviously is closed in N n fpg. ThusDp�BN �p; q�� is closed inDp�N n fpg� and of Lebesgue measure 0 by Fubini's theorem.Thus Dp�BN �p; q�� is Jordan measurable and of measure 0. The latter implies thatBN�p; q� is Jordan measurable in N and of measure 0, concluding the proof of (6.1).For an alternative proof of (6.1) use our assumption about M to show that Mis an Alexandrov space and the fact that in Alexandrov spaces bisectors havemeasure 0. The latter is a consequence of a result of Shiohama and Tanaka [30].

316 PETER M. GRUBER

Let Nn � fp1; . . . ; png � J. If Nn is su¤ciently dense in the interior of J; intJ,then the Dirichlet-Voronoi cells

Di � DM�Nn; pi� � fx 2 J : RM�pi; x�W RM�pj; x� for each pj 2 Nng

are Jordan measurable sets which tile J. (6.2)

Consider a ¢nite cover of cl J by neighborhoods as described in (6.1). Then, if Nn issuf¢ciently dense in intJ, each Di and all points pj 2 Nn which are needed to de¢neit, are contained in one of these neighborhoods, say N. The boundary bdDi thenconsists of subsets of bisectors BN �pi; pj� and of subsets of bd J. By (6.1) and sinceJ is Jordan measurable, it follows that each Di has boundary of measure 0 and thusis Jordan measurable. In addition, the overlap with any other Dj (is empty or)has measure 0. The proof of (6.2) is complete.

Let Nn � fp1; . . . ; png � J be su¤ciently dense in intJ and letWn � fw1; . . . ;wng � R. Then

E�Hf ;Nn;Wn�XE�Hf ;Nn;Wn�

�ZJ

minp2Nnff �RM�p; x��gdoM�x�;

where Wn � �w1; . . . ;wn� with wi � oM�Di�. (6.3)

We assume that Nn is so dense in intJ, that (6.2) holds. Then the de¢nition ofE�Hf ;Nn;Wn�, proposition (6.2), the fact that the functionminff �RM�pi; x�� : pi 2 Nng is in the class Hf , is 0 at the points pi, and the de¢nitionof E�Hf ;Nn;Wn� together imply that

E�Hf ;Nn;Wn� � supg2HffjZJg�x� doM�x� ÿ

Xni�1

wig�pi�jg

W supg2HffXni�1

ZDi

jg�x� ÿ g�pi�jdoM�x�gWXni�1

ZDi

f �RM�pi; x�� doM�x�

�ZJ

minpi2Nnff �RM�pi; x��g doM�x�WE�Hf ;Nn;Wn�: (6.4)

This con¢rms the inequality in (6.3). Putting Wn �Wn in (6.4), equality holdsthroughout, which yields the equality in (6.3).

6.2.2. For the proof of Theorem 5(i) note that E�Hf ; n� ! 0 as n!1. Thus for allsuf¢ciently large n it is suf¢cient to consider in the de¢nition of E�Hf ; n� only such


sets of nodes Nn which are suf¢ciently dense in intJ in the sense of proposition (6.3).Then

E�Hf ; n� � infNn;Wn

E�Hf ;Nn;Wn�� infNn

E�Hf ;Nn;Wn�

� infNn

ZJ

minp2Nnff �RM�p; x��

� �doM�x�g

for all su¤ciently large n.

Now apply Theorem 1.

6.2.3. For the proof of Theorem 5(ii) the fact that E�Hf ; n� ! 0 as n!1 and theassumption of part (ii) imply that Nn is suf¢ciently dense in intJ in the sense of (6.3)for all suf¢ciently large n. Proposition (6.3) and the de¢nition of E�Hf ; n� then showthat

E�Hf ; n� � E�Hf ;Nn;Wn�XE�Hf ;Nn;Wn�

�ZJ

minp2Nnff �RM�p; x��g doM�x� � E�Hf ;Nn;Wn�XE�Hf ; n�

for all suf¢ciently large n. This, in turn, implies that,ZJ

minp2Nnff �RM�p; x��gdoM�x� � E�Hf ; n�:

Combining this with (i) and Theorem 2 yields (ii) and thus concludes the proof ofTheorem 5. &

Acknowledgements

The work on this article was begun during a visit to the MSRI in Berkeley in April1996. For the invitation many thanks are due to Professors Milman and Thurston.For valuable hints I am obliged to Professors Goodey and Schnitzer and to DrWallner. I am grateful to Professor Zam¢rescu who informed me about the neededconsequence of the work of Shiohama and Tanaka.

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optimal configurations of finite sets in riemannian 2-manifolds

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