On the regularity of the distribution of the Fekete points of a compact surface in R"
~)ETER SJ6GREN University of G6teborg, Sweden
1. I n t r o d u c t i o n
The question to be considered in this note can be stated in terms of classical physics. L e t a finite number of equal point charges find a stable equilibrium distri- bution within an isolated conductor. Then how well does this distribution approximate the equilibrium distribution of the same total charge?
We let S be a compact ( n - 1)-dimensional surface of class C 1'~ in R", which is supposed to be the common boundary of two domains. Only the case n >_ 3 will be considered, although the method of proof works also in the plane for a simple closed curve of the same regularity. For N > 1 we let (z•l, 9 9 Z~xN) be a system of Fekete points of S, i.e., an N-tuple of points in S which among all N-tuples of points in S minimizes the energy
1 1
N2 ~ 2,j lzNi zrvj
9 . - - I r t - 2of the mass distribution /~N consisting of point, masses
1/N
at each z~7 i. I-Iere we omit the infinite energy of each point mass. The ~'ekete points need not be uniquely determined by S and N.L e t ~ be the eqmlibrium distribution of total mass 1 in S, to which #N will be compared. I n two dimensions several estimates of # N - 2 have been made.
I n t h a t case our method yields the result t h a t a n y subarc B of the curve satisfies
%/log N I/*N(B) -- 2(B) I _< const. - % / ~
This should be compared to Kleiner's bound, eonst, log N / % / N in [1]. ~'or analytic curves Pommerenke has given l~nch sharper estimates in for example [3], where further references can be found.
148 I'E TEl~ SJ()G:REN
:Kleiner's paper [2] contains as a special case of an auxiliary result an estimate of this kind in more t h a n two dimensions. He assumes t h a t the surface S is of class C ~ a n d chooses a fixed subdivision $ 1 , . . . , S~ of S such t h a t each Si is the image under a C 2 mapping ~ of a unit square. As test sets B Kleiner takes the images under every ~i of all axial subreetangles of the unit square, and he obtains the estimate
1
I/~(B) -- 2(B)[ ~ const. N 3(~-1)
We consider a larger class of test sets t h a n Kleiner does, a n d our estimate is sharper.
However, at least for more regular surfaces, it is probably not best possible.
The author wishes to t h a n k Prof. J. Korevaar for fruitful discussions on the subject of Fekete points.
2. Formulation of the theorem
To define test subsets of
S,
we need the concept of K-regular boundary, which is defined in SjSgren [4], as follows.Definition.
A set B C S is said to have K-regular b o u n d a r y (K ~ 0) if for a n y d ~ 0f dS ~ Kd, B*d
where
z* = {x e s: e(x, B*) < d}.
Here B* is the b o u n d a r y of B in the relative topology of S, while ~ means distance in R n. The measure
dS
is the area measure of S.T~EOI~EM.
For any B c S having K-regular boundary
1 1
[#~(B) -- 2(B)[ ~
CK2N 2(,,-x), C -= C(S).
~emarl~
1. As can be seen geometrically, it is to be expected t h a t the exponent of hr depends on the dimension.lCemarl~
2. I f we let B be the intersection of S and suitable balls with centres in S, we can deduce certain estimates of distances from a point in S to the closest Fekete point. For example, the distance d from a Fekete point zN~ to its closest neighbourZlvi
satisfiesd _< CN-~, C = C(S),
where
O N T H E I ~ E G U L A R I T Y " O F T H E D I S T R I B U T I O N O F T H E F E K E T E P O I N T S 149
1
Y - - 2(n - - 1) ~"
3. P r o o f of the t h e o r e m
F o r measures /~ and v with compact support in m~ttual energy, and energy norm by, respectively,
and
R ~, we define the potential,
f d/~(y) U.(x) = Ix _
y [ , , - 2 ,ll~II = (~, ~)89
The letter C will denote various constants, all of which depend only on S.
We let EN be the finite part of the energy of /tN:
1 1
E ~ - N2 ~ lz~i --zNjl n-~
and start b y estimating the potential U"N on S. I f
fi(z) =jZ lz - z~j? -2
1the minimum p r o p e r t y of {ziv~} implies that
f,(z) • fi(zNi)
if Z e S. Adding these inequalities, we get
N ( N - 1)U'~(z) > N2EN. (1)
Using some ideas from Kleiner [2], we shall estimate EN in terms of the energy of 2, which we call E and which is also the value on S of the potential U x.
Let vN~ be the uniform distribution of the mass
1/N
on the sphere ]z ~ z~[ = r, where r > 0 will be determined later. We p u ti
Since
1 1
U ~ ( z ) < _~ [z - z~,l ~-2
1 5 0 : P E T E R S J 6 G I ~ E N
for all z, we see t h a t for i # j
A d d i n g , we g e t
1 1
> \
5 ~ l z ~ - z~j]~-~ ~'~)'
EN >_ 11~I[ ~ - ~ II~NfL ~ = I!~.~[l ~ 2Vr~_ ~ (2)
i
I t follows f r o m T h e o r e m 2.4 in W i d m a n [5] t h a t t h e g r a d i e n t o f U ~ is b o u n d e d n e a r S, so t h a t
I(~lv, ~) - - E l =
f ( u ~ -
E ) d v N<
Cr.T h e r e f o r e
llvNll ~ - - - - 11211 z + 2(,,N, 2) -~ ]lvN - - 2112 >~ E - - Cr. (3) N o w (1), (2), and (3) i m p l y
U ' N > E - - Cr - - ( N - 1)r n-2 o n S. I f we t a k e
we see t h a t
1
1
s u p U~--I'N < C N ~ - 1 ,
S
a n d t h e t h e o r e m follows f r o m SjSgren [4, T h e o r e m 1].
References
1. KLEINER, W., Sur l ' a p p r o x i m a t i o n de la reprdsentation conforme par la mdthode des points extrdmaux de M. Leja. A n n . Pol. M a t h . 14 (1964), 131--140.
2. --~>-- Degree of convergence of the extremal points m e t h o d for Dirichlet's problem in the space. Coll. M a t h . 12 (1964), 41--52.
3. PO~EREXKE, Cm, Uber die Verteilung der F e k e t e - P u n k t e I I . M a t h . A n n . 179 (1969), 212--218.
O N T H E ] ~ E G U L A I : C I T u O F T H E D I S T I : ~ I B U T I O ~ O F T H E F E K E T E P O I N T S ] 5 1
4. S J O G I ~ E N , P . , Estimates of mass distributions from their potentials a n d energies. A r k . ~]Jat.
10 (1972), 59 77.
5. WID~AN, K.-O., inequalities for the Green function and b o u n d a r y c o n t i n u i t y of the gradient of solutions of elliptic differential equations. M a t h . S c a n d . 21 (1967), 17--37.
Received M a r c h 17. 1972 Peter Sj6gren
D e p a r t m e n t of Mathematics University of GSteborg Faek
S-402 20 GS~eborg Sweden
